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The dataset generation failed
Error code: DatasetGenerationError
Exception: ArrowInvalid
Message: JSON parse error: Missing a closing quotation mark in string. in row 36
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 145, in _generate_tables
dataset = json.load(f)
File "/usr/local/lib/python3.9/json/__init__.py", line 293, in load
return loads(fp.read(),
File "/usr/local/lib/python3.9/json/__init__.py", line 346, in loads
return _default_decoder.decode(s)
File "/usr/local/lib/python3.9/json/decoder.py", line 340, in decode
raise JSONDecodeError("Extra data", s, end)
json.decoder.JSONDecodeError: Extra data: line 2 column 1 (char 71776)
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1995, in _prepare_split_single
for _, table in generator:
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 148, in _generate_tables
raise e
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 122, in _generate_tables
pa_table = paj.read_json(
File "pyarrow/_json.pyx", line 308, in pyarrow._json.read_json
File "pyarrow/error.pxi", line 154, in pyarrow.lib.pyarrow_internal_check_status
File "pyarrow/error.pxi", line 91, in pyarrow.lib.check_status
pyarrow.lib.ArrowInvalid: JSON parse error: Missing a closing quotation mark in string. in row 36
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1529, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1154, in convert_to_parquet
builder.download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1027, in download_and_prepare
self._download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1122, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1882, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 2038, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
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text
string | meta
dict |
|---|---|
\section{Introduction}
Cosmology is one of the most exciting adventures in the human endeavour: the understanding of the origin, evolution and future of our Universe by combining the physics at micro- and macro-scales in a joint framework. In particular, the Universe acceleration is one of the most important puzzles. Discovered by the Supernovae team, through type Ia supernovae \citep[SNIa,][]{Riess:1998}, it is also confirmed by the acoustic peaks of the Cosmic Microwave Background Radiation
\citep[CMB,][]{WMAP:2003elm}, and recently tested with large-scale structure measurements \citep{Nadathur:2020kvq}. Tackling the Universe acceleration is indeed a complicated issue due to the attractive nature of gravity in the General Relativity (GR) framework.
The first approach on its explanation is through the addition of the cosmological constant
\citep[CC,][]{Carroll:2000}. Thus, using the continuity equation and assuming a constant energy density ($\dot{\rho}=0$), it is possible to conclude that the equation of state (EoS) is $w=-1$, which is in concordance with what is expected for a fluid that accelerates the Universe. Another main characteristic is that the energy density of the CC must be subdominant in order to obtain a late and non-violent acceleration. From the Quantum Field Theory (QFT) viewpoint, the CC can be explained by the addition of
quantum vacuum fluctuations (QVF) associated to the space-time. Therefore, the expansion of space-time implies an increase of QVF, maintaining a constant energy density. However, when we calculate the energy density of the QVF, the result is in complete disagreement with the observed one \citep[see][]{Zeldovich:1968ehl,Weinberg}, this is the so-called {\it fine tuning problem} \citep{Addazi:2021xuf}.
In addition to the cosmological constant problem, a $4.2\sigma$ tension in the current Hubble parameter value ($H_0$) measured by Supernova $H_0$ for the Equation of State (SH0ES) collaboration \citep{Riess:2020fzl} and the one obtained by Planck collaboration under the $\Lambda$-Cold Dark Matter ($\Lambda$CDM) scenario \citep{Planck:2018vyg} has been recently observed.
The above mentioned problems that afflicts the understanding of the CC in this framework has driven the community to propose other approaches like scalar fields, dynamical dark energy, viscous fluids, Chaplygin gas \citep{Hernandez-Almada:2018osh} or modifications to GR such as unimodular gravity \citep{Garcia-Aspeitia:2019yni,Garcia-Aspeitia:2019yod}, Einstein-Gauss-Bonet \citep{Garcia-Aspeitia:2020uwq}, Brane Worlds \citep{Garcia-Aspeitia:2018fvw}, among others
\citep[see][for a compilation of the mentioned previous models]{CANTATA:2021ktz,Motta:2021hvl}.
Despite the fact that we can have numerous models and scenarios describing the late time acceleration, at the end of the day the detailed confrontation with observations, alongside theoretical consistency, will be the main method for
their validation.
One interesting alternative to investigate the dynamics of the Universe is the gravity-thermodynamics approach \citep{Jacobson:1995ab,Padmanabhan:2003gd,Padmanabhan:2009vy}. This takes advantage of the first law of thermodynamics and the standard Bekenstein-Hawking entropy-area relation for black holes, which applied to the apparent cosmological horizon leads to the Friedmann equations \citep{Frolov:2002va,Cai:2005ra,Akbar:2006kj,Cai:2006rs}. Hence, the application of this conjecture with various alternative entropy relations leads to modified Friedmann equations whose additional terms can source the cosmic acceleration \citep{Cai:2006rs,Akbar:2006er,Paranjape:2006ca,Sheykhi:2007zp,Jamil:2009eb,Cai:2009ph,Wang:2009zv,Jamil:2010di,Sheykhi:2010wm,Sheykhi:2010zz,Gim:2014nba,Fan:2014ala,Lymperis:2018iuz,Sheykhi:2018dpn,Saridakis:2020lrg}.
One interesting class of extended entropies arises through
generalizations for the Boltzmann-Gibbs statistics. In particular, one modifies
the classical entropy of a system given by $S=-k_{B} \sum_{i}^{} p_{i} \ln{p_{i}}$, where $p_{i}$ is the probability of a system to be within a microstate, through non-extensive
analysis
resulting into the Tsallis entropy \citep{Tsallis:1988, Lyra:1998}, through quantum-gravitational considerations resulting into Barrow entropy \citep{Barrow:2020tzx}, or through relativistic extensions resulting into Kaniadakis entropy \citep{Kaniadakis:2002zz,Kaniadakis:2005zk}.
Hence, the application of the gravity-thermodynamics conjecture using the above modified horizon entropy gives rise to modified cosmological scenarios. In \citet{Lymperis:2018iuz} this was performed in the framework of Tsallis entropy, in
\citet{Saridakis:2020lrg,Saridakis:2020cqq,Leon:2021wyx,Barrow:2020kug}
applied it for the Barrow entropy case, and recently
\citep{Lymperis:2021qty} use it in the Kaniadakis entropy frame.
In the present work, we
investigate Kaniadakis horizon entropy cosmology by confronting it to observational data from Cosmic Chronometers, Supernovae Type I (SNIa), HII galaxies, strong lensing systems (SLS), and baryon acoustic oscillations (BAO) observations. Additionally, we
perform a complete dynamical system analysis in order to etxract information of the local and global features of the cosmological evolution.
The outline of the paper is as follows: Sec. \ref{sec:cosmo} introduces the framework of the Kaniadakis cosmology. In Sec. \ref{sec:data} we infer the cosmological parameter under the Kaniadakis model using five observational datasets mentioned above.
The section \ref{sec:SA}
presents a stability analysis around the equilibrium points of the dynamical system under the Kaniadakis cosmology. Finally, we discuss and summarize our results in Section \ref{sec:Con}. From now on we use natural units in which $\hbar=k_B=c=1$.
\section{Kaniadakis horizon entropy cosmology} \label{sec:cosmo}
In this section we present the scenario of Kaniadakis horizon entropy
cosmology \citep{Lymperis:2021qty}, namely the modified Friedmann equations
arising from the application of the gravity-thermodynamics conjecture using the
extended Kaniadakis entropy.
Kaniadakis entropy is an one-parameter
generalization of the classical entropy, given as
$S_{K}=- k_{_B} \sum_i n_i\, \ln_{_{\{{\scriptstyle
K}\}}}\!n_i $ \citep{Kaniadakis:2002zz,Kaniadakis:2005zk},
with $k_{_B}$ the Boltzmann constant and
where $\ln_{_{\{{\scriptstyle
K}\}}}\!x=(x^{K}-x^{-K})/2K$. In such a framework the
dimensionless parameter $-1<K<1$ quantifies the relativistic
deviations from standard statistical mechanics, and the latter is recovered in
the limit $K\rightarrow0$.
Kaniadakis entropy can be expressed as
\citep{Abreu:2016avj,Abreu:2017hiy,Abreu:2021avp}
\begin{equation}
\label{kstat}
S_{K} =-k_{_B}\sum^{W}_{i=1}\frac{P^{1+K}_{i}-P^{1-K}_{i}}{2K},
\end{equation}
with $P_i$ the probability of a specific microstate and
$W$ the total configuration number. When we apply it in the
black-hole framework, we obtain
\citep{Drepanou:2021jiv,Moradpour:2020dfm,Lymperis:2021qty}
\begin{equation} \label{kentropy}
S_{K} = \frac{1}{K}\sinh{(K S_{BH})},
\end{equation}
where
\begin{equation}
\label{Horentropy}
S_{BH}=\frac{1}{4G} A,
\end{equation}
is the usual Bekenstein-Hawking entropy, with $A$ the horizon
area and $G$ is the gravitational constant.
Hence, in the limit $K\rightarrow 0$ we
recover Bekenstein-Hawking entropy.
Let is now apply the gravity-thermodynamics conjecture using Kaniadakis
entropy. We consider a flat
homogeneous and isotropic Friedmann-Robertson-Walker (FRW) metric
of the form
\begin{equation}
ds^2=-dt^2+a^2(t)\left(dr^2+r^2d\Omega^2 \right),
\label{metric}
\end{equation}
where $d\Omega^2\equiv d\theta^2+\sin^2\theta d\varphi^2$ is the solid angle,
and $a(t)$ the scale factor.
In this setup,
the
first law of thermodynamics is interpreted in terms of the heat/energy that flows through the apparent horizon of the Universe
\citep{Jacobson:1995ab,Padmanabhan:2003gd,Padmanabhan:2009vy}, which in the case
of flat geometry is just
\citep{Bak:1999hd,Frolov:2002va,Cai:2005ra,Cai:2008gw}:
\begin{equation}
\label{apphor}
r_a=\frac{1}{H },
\end{equation}
where $H=\dot a/a$ is the Hubble parameter (dots denote derivatives
with respect to $t$).
Concerning the horizon temperature,
this is given by the standard
relation \citep{Gibbons:1977mu}:
\begin{equation}
\label{Th}
T=\frac{1}{2\pi r_a}.
\end{equation}
Hence, the
first law of thermodynamics is just
$-dE=TdS$, where $-dE=A(\rho_m+p_m)H r_{a}dt$ is
the energy flow through the horizon during a time
interval $dt$ in the case of a Universe filled with a matter perfect fluid
with energy density $\rho_m$ and pressure $p_m$ \citep{Cai:2005ra}.
Differentiating (\ref{kentropy}) we obtain
\begin{equation}
\label{dsk}
dS_{K}=\frac{8\pi}{4G}\cosh{\left(K \frac{\pi }{GH^2}
\right)}r_{a}\dot{r}_{a}dt,
\end{equation}
where we have used that $A=4\pi r_{a}^2=4 \pi/H^2 $.
Inserting (\ref{Horentropy}), (\ref{Th}), and
(\ref{dsk}) into the first
law of thermodynamics, alongside with the relation $\dot r_{a}=-\dot{H}/H^2$,
we acquire \citep{Lymperis:2021qty}
\begin{equation} \label{gfe1}
-4\pi G(\rho_{m}+p_{m})=\dot{H} \cosh{\left[K
\frac{\pi}{GH^2}\right]}.
\end{equation}
Thus, using the matter conservation equation
\begin{equation}
\dot{\rho}_m +3H(\rho_m +p_m)=0,
\label{matterconsv}
\end{equation}
the integration of (\ref{gfe1}) leads to
\begin{eqnarray} \label{gfe2}
&&
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\frac{8\pi G}{3}\rho_{m}= H^{2} \cosh{\left[K
\frac{\pi}{GH^2 }\right]}
-\frac{K\pi}{G} \text{shi}{\left[K
\frac{\pi}{GH^2 }\right]}-\frac{\Lambda}{3},
\end{eqnarray}
with $\Lambda$ the integration constant and ${\rm
shi}(x)\equiv\int_0^{x}\sinh(x')/x'dx'$, a mathematical odd function of $x$ with
no discontinuities.
The above equations (\ref{gfe1}) and (\ref{gfe2}) are the modified Friedmann
equations in the scenario of Kaniadakis horizon entropy cosmology
\citep{Lymperis:2021qty}.
As expected, in the limit
$K\rightarrow 0$ they turn into the standard ones. We can re-write them as
\begin{align}
H^2 & =\frac{8\pi G}{3}(\rho_m+\rho_{DE}), \label{H1}
\\
\dot{H}& =-4\pi G(\rho_m+\rho_{DE}+p_m+p_{DE}), \label{H2}
\end{align}
where we have introduced an effective dark-energy sector, with energy density
and pressure respectively of the form
\begin{small}
\begin{align}
&\rho_{DE}=\frac{3}{8\pi G}\Bigg\lbrace\frac{\Lambda}{3}+H^2\left[1-\cosh\left(\frac{\pi K}{GH^2}\right)\right] +\frac{\pi K}{G}{\rm shi}\left(\frac{ \pi K}{GH^2}\right)\Bigg\rbrace, \\
&p_{DE}=-\frac{1}{8\pi
G}\Bigg\lbrace\Lambda+(3H^2+2\dot{H})\left[1-\cosh\left(\frac{\pi
K}{GH^2}\right)\right] +\frac{3\pi K}{G}{\rm shi}\left(\frac{\pi
K}{GH^2}\right)\Bigg\rbrace \label{PDE}.
\end{align}
\end{small}
Therefore, the EoS parameter for the dark energy
sector is
\begin{align}
w_{DE}= & -1-2\dot{H}\left[1\!-\!\cosh\left(\frac{\pi
K}{GH^2}\right)\!\right] \times \nonumber \\
& \left\lbrace\Lambda+3H^2\left[1\!-\!\cosh\left(\frac{
\pi K}{GH^2}\right)\! +\!\frac{3\pi K}{G}{\rm shi}\left(\frac{\pi
K}{GH^2}\right)\right]\right\rbrace^{-1}.
\end{align}
In the general case, the cosmological equations are the two Friedmann equations
\eqref{H1} and \eqref{H2}, alongside the matter conservation equation
(\ref{matterconsv}). \textbf{For convenience we focus on on the dust case,
namely we consider $p_m=0$}. It proves convenient to express the
equations in terms of
dimensionless variables.
Introducing the density parameters
\begin{align}
\Omega_{\Lambda}\equiv \frac{\Lambda}{3 H^2},
\quad \Omega_{m}\equiv \frac{8 \pi G \rho_m}{3 H^2},
\end{align}
the normalized Hubble function
\begin{align}
E\equiv \frac{H}{H_0},
\end{align}
with $H_0$ the Hubble parameter at the present scale factor $a_0$, and
defining the
dimensionless parameter $\beta$ as
\begin{align}
\beta\equiv \frac{K \pi }{G H_0^2},
\end{align}
then, the cosmological equations are expressed as
\begin{align}
& \Omega_\Lambda^{\prime}(N)= 3 \Omega_\Lambda \Omega_m
\text{sech}\left(\frac{\beta }{E^2}\right),\\
& \Omega_m^{\prime}(N)= 3
\Omega_m \left[\Omega_m \text{sech}\left(\frac{\beta
}{E^2}\right)-1\right], \\& E^{\prime}(N)= -\frac{3}{2} E
\Omega_m \text{sech}\left(\frac{\beta
}{E^2}\right),
\end{align}
where primes denote derivatives with respect to the e-foldings
number $N= \ln (a/a_0)$ (and thus $ f^{\prime} =
\dot{f}/H$). Note that using the above variables, the first Friedmann equation
\eqref{H1} gives rise to the constraint
\begin{align}
\beta \text{shi}\left(\frac{\beta }{E^2}\right)+E^2
\left[-\cosh \left(\frac{\beta }{E^2}\right)+\Omega_\Lambda
+ \Omega_{m}\right]=0,\label{eqN4.9}
\end{align}
which allows us to eliminate $\Omega_\Lambda$ in terms of $\Omega_{m}$ and $E$.
Finally, note that for the effective dark energy density parameter, in the
general case we have
\begin{equation}
\Omega_{DE}= 1-\Omega_m = 1 + \beta \text{shi}\left( \beta
\right) -\cosh \left( \beta \right)+ \Omega_\Lambda .
\label{constreq1}
\end{equation}
Lastly, it proves convenient to introduce the
deceleration parameter $q(z)$, and the cosmographic
jerk parameter $j(z)$, which are defined as
\begin{align}
q :=& -1- \frac{E^{\prime}}{E}, \label{q}\\
j:= & q(2q+1)-q', \label{j}
\end{align}
where $j=1$ recovers the case of a cosmological constant.
In the scenario of Kaniadakis horizon entropy cosmology one may have
the general integration constant $\Lambda$, which will play the role of an
explicit cosmological constant, or one may set it to zero and thus require for
the extra $K$-dependent terms to drive the Universe acceleration. Since the
corresponding equation structure (which will be used later for the the
Bayesian statistical analysis and the dynamical
system approach) is different in the two cases, we
examine them separately in the following subsections.
\subsection{Case I: $\Lambda \neq 0$}
In the general case where $\Lambda \neq 0$, i.e. $\Omega_\Lambda\neq 0$, we
use the constraint equation (\ref{eqN4.9}) to obtain the reduced dynamical
system
\begin{align}
& \Omega_m^{\prime}(N)= 3
\Omega_m \left[\Omega_m \text{sech}\left(\frac{\beta
}{E^2}\right)-1\right], \label{evolm}\\& E^{\prime}(N)= -\frac{3}{2} E
\Omega_m \text{sech}\left(\frac{\beta
}{E^2}\right). \label{evolE}
\end{align}
This is integrable, with
\begin{small}
\begin{align}
& \Omega_{m}(E)= \cosh \left(\frac{\beta
}{E^2}\right)+\frac{-\cosh (\beta )-\beta
\text{shi}\left(\frac{\beta }{E^2}\right)+\beta
\text{shi}(\beta )+ \Omega_m^{(0)}}{E^2}, \\
& \Omega_m(1)=\Omega_{m}^{(0)},
\end{align}
\end{small}
and
\begin{small}
\begin{align}
& E^{\prime}(N) = -\frac{3}{2} E -\frac{3 \text{sech}\left(\frac{\beta }{E^2}\right)
\left(-\cosh (\beta )-\beta \text{shi}\left(\frac{\beta
}{E^2}\right)+\beta \text{shi}(\beta )+ \Omega_m^{(0)}\right)}{2 E}, \label{4.13}
\end{align}
\end{small}
where $\Omega_m(N=0)=\Omega_{m}^{(0)}$ and $E(N=0)=1$.
The equation
\eqref{4.13} is easily integrated to give
\begin{small}
\begin{align}
{3} N (E)= & -\ln
\Bigg( 1-\frac{\cosh (\beta )-E^2 \cosh \left(\frac{\beta
}{E^2}\right)+\beta \text{shi}\left(\frac{\beta
}{E^2}\right)-\beta \text{shi}(\beta )}{\Omega_m^{(0)}}\Bigg),
\end{align}
\end{small}
introducing as dynamical variable the redshift
$ e^N= a= (1+z)^{-1}$, where $z=0$ and $a_0=1$ for current time, the previous equation leads to
\begin{small}
\begin{align}
& (z+1)^{{3 }}= 1-\frac{\left[\cosh (\beta )-E^2 \cosh \left(\frac{\beta
}{E^2}\right)+\beta \text{shi}\left(\frac{\beta
}{E^2}\right)-\beta \text{shi}(\beta )\right]}{\Omega_m^{(0)}}. \label{eqN4.17}
\end{align}
\end{small}
Evaluating \eqref{constreq1} at present time
gives
\begin{equation}
\Omega_{DE}^{(0)}= 1-\Omega_m^{(0)}= 1 + \beta \text{shi}\left( \beta
\right) -\cosh \left( \beta \right)+ \Omega_\Lambda^{(0)},
\label{exact3.20aa}
\end{equation}
and combining it with \eqref{eqN4.17} we have
\begin{align}
& \Omega_m^{(0)} (z+1)^{3}+ \Omega_\Lambda^{(0)}=E^2 \cosh \left(\frac{\beta
}{E^2}\right) - \beta \text{shi}\left(\frac{\beta
}{E^2}\right). \label{exact3.20}
\end{align}
We mention here that in the general case where $\Lambda\neq 0$ from the above
we obtain the relation between $\beta$, $ \Omega_m^{(0)}$ and $
\Omega_\Lambda^{(0)}$ as
\begin{equation}
\label{exact3.20aan}
\Omega_m^{(0)}= \cosh \left( \beta \right) - \beta \text{shi}\left( \beta
\right) - \Omega_\Lambda^{(0)}.
\end{equation}
We expand (\ref{exact3.20aa}), (\ref{exact3.20})
up to third order around $\beta=0$, resulting in
\begin{equation}
\Omega_m^{(0)}+\Omega_\Lambda^{(0)}=1-\frac{\beta ^2}{2}+O\left(\beta
^4\right),
\end{equation}
and
\begin{align}
\Omega_m^{(0)} (z+1)^{{3 }}+ \Omega_\Lambda^{(0)}& \approx E^2-\frac{\beta
^2}{2 E^2}+O\left(\beta ^4\right).
\end{align}
Hence, we obtain four roots:
\begin{small}
\begin{align}
&E_{1,2}= \mp\frac{\sqrt{\Omega_m^{(0)} (z+1)^{{3 }}+
\Omega_\Lambda^{(0)}-\sqrt{2 \beta ^2+\left[\Omega_m^{(0)} (z+1)^{{3 }}+
\Omega_\Lambda^{(0)}\right]^{2}}}}{\sqrt{2}}, \\
&E_{3,4}=\mp\frac{\sqrt{\Omega_m^{(0)} (z+1)^{{3 }}+
\Omega_\Lambda^{(0)}+\sqrt{2 \beta ^2+\left[\Omega_m^{(0)} (z+1)^{{3 }}+
\Omega_\Lambda^{(0)}\right]^{2}}}}{\sqrt{2}}. \label{E4}
\end{align}
\end{small}
Solutions $E_1$ and $E_2$ are complex, and $E_3$ is negative. Thus,
the only physical solution is $E_4$.
In the following, instead of using the exact implicit formula for $E$
given in \eqref{exact3.20}, we will consider the approximation $E_4$ in
\eqref{E4}.
\subsection{Case II: $\Lambda = 0$}
In the case where an explicit cosmological constant is absent, namely $\Lambda
= 0$, i.e $\Omega_\Lambda=0$, the general system \eqref{evolm}, \eqref{evolE}
reduces to
\begin{equation}
E^{\prime}(N)=-\frac{3}{2} E + \frac{3 \beta \text{sech}\left(\frac{\beta
}{E^2}\right) \text{shi}\left(\frac{\beta
}{E^2}\right)}{2 E}.
\end{equation}
The last equation is easily integrated to give
\begin{align}
N (E)= -\frac{1}{3}\ln \left[\frac{E^2 \cosh \left(\frac{\beta }{E^2}\right)-\beta \text{shi}\left(\frac{\beta }{E^2}\right)}{\cosh (\beta )-\beta \text{shi}(\beta )}\right],
\end{align}
which, using the redshift,
implies
\begin{align}
& (z+1)^{{3 }}=\frac{E^2 \cosh \left(\frac{\beta }{E^2}\right)-\beta
\text{shi}\left(\frac{\beta } {E^2}\right)}{\cosh (\beta )-\beta
\text{shi}(\beta )}. \label{eqN20}
\end{align}
Hence, using \begin{equation}
\label{corr}
\Omega_m^{(0)}= {\cosh \left( \beta \right)-\beta \text{shi}\left(\beta\right)},
\end{equation}
as it arises from \eqref{exact3.20aan} for $\Omega_\Lambda^{(0)}=0$,
we obtain
\begin{align}
& \Omega_m^{(0)} (z+1)^{{3 }}= {E^2 \cosh \left(\frac{\beta }{E^2}\right)-\beta \text{shi}\left(\frac{\beta }{E^2}\right)}. \label{eqN3.34}
\end{align}
Expanding (\ref{eqN3.34})
up to third order around $\beta=0$, we result to
\begin{align}
\Omega_m^{(0)} (z+1)^{{3 }} & \approx E^2-\frac{\beta ^2}{2
E^2}+{\mathcal{O}}\left(\beta ^4\right),
\end{align}
and thus at present times gives $
\Omega_m^{(0)}= 1-\frac{\beta ^2}{2}+{\mathcal{O}}\left(\beta ^4\right)$.
Therefore, we obtain four roots:
\begin{small}
\begin{eqnarray}
&& E_{1,2}= \mp\frac{\sqrt{{\Omega_m^{(0)}} (z+1)^{ 3}-\sqrt{2 \beta
^2+{\Omega_m^{(0)}}^2 (z+1)^{6}}}}{\sqrt{2}},\\
&&E_{3,4}=\mp\frac{\sqrt{{\Omega_m^{(0)}} (z+1)^{ 3}+\sqrt{2 \beta
^2+{\Omega_m^{(0)}}^2 (z+1)^{6}}}}{\sqrt{2}}. \label{eqN340}
\end{eqnarray}
\end{small}
Similarly to the previous case, roots $E_1$ and $E_2$ are complex
while $E_3$ is negative.
Consequently, the only physical solution is $E_4$ in \eqref{eqN340}.
\section{Observational constraints} \label{sec:data}
In this section we confront the scenario of Kaniadakis horizon-entropy
cosmology with observations. We are interested in extracting the bounds on the
parameter phase-space $\Theta = \{h, \Omega_m^{(0)}, \beta\}$ and $\{h,
\beta\}$,
particularly on the parameter $\beta$, which is related to the
Kaniadakis basic parameter $K$. For
convenience, we focus on the physically interested case of dust matter, namely
we set $w_m=0$.
\subsection{Datasets and methodology} \label{subsec:data}
We will employ the most commonly used datasets.
\begin{itemize}
\item {\it Observational Hubble Data} (OHD).
The sample contains $31$ cosmological-independent measurements of the Hubble parameter in the redshift range $0.07<z<1.965$ from passive elliptic galaxies, the so-called cosmic chronometers \citep{Moresco:2016mzx}.
\item {\it Pantheon Supernova Type Ia sample} (SNIa). We use 1048
data points of the distance modulus, $\mu(z)_{\mathrm{SNIa}}$, of high-redshift
SNIa in the redshift range $0.001<z<2.3$ \citep{Scolnic:2018}.
\item {\it HII galaxies} (HIIG).
It contains a total of $181$ data points of the distance modulus $\mu_{\mathrm{HIIG}}(z)$ estimated from the Balmer line luminosity-velocity dispersion relation for HII galaxies spanning the redshift region $0.01<z<2.6$ \citep{Gonzalez-Moran:2021drc}.
\item {\it Strong lensing systems} (SLS). We use the sample by \citet{Amante_2020} which contains $143$ strong lensing systems by elliptical galaxies with measurements of the redshift for the lens and the source, spectroscopic velocity dispersion and the Einstein radius.
These quantities
allow us to construct an observational distance ratio within the region $ 0.5 \leq D^{obs} \leq 1$ .
\item {\it Baryon acoustic oscillations} (BAO). We consider $6$ correlated data points of the imprint of baryon acoustic oscillation in the size of the sound horizon in clustering and power spectrum of galaxies measured by \citet{Percival:2010,Blake:2011,Beutler:2011hx} and collected by \citet{Giostri:2012}.
\end{itemize}
We would like to mention here that other
cosmological observations could be included in the parameter estimation too,
for instance the CMB data. To perform such analysis in a robust
way, a full perturbation approach is needed in order to obtain the linear
Einstein-Boltzmann equations. Nevertheless this is beyond the scope of the
present work.
An alternative approach would be to use the distance priors from Planck 2018
based on slight deviations from $\Lambda$CDM, such as the $w$CDM model
\citep{Chen:2019JCAP}. However, since this procedure could lead to biased
constraints, in the following we prefer not to use the CMB dataset.
The inference of the cosmological parameters under Kaniadakis horizon entropy
cosmology for both scenarios ($\Lambda\neq0$ and $\Lambda=0$)
is performed by a Bayesian Markov Chain Monte Carlo (MCMC) approach and the \texttt{emcee} Python module \citep{Emcee:2013}. We set $3000$ chains with $250$ steps each, and consider uniform priors in the ranges: $h:[0.2, 1]$, $\Omega_m^{(0)}:[0,1]$, $\beta: [-\pi, \pi]$. The burn-in phase is stopped up to obtain convergence according to the auto-correlation time criterion.
Then, we build a Gaussian
log-likelihood as the merit-of-function to minimize
through the equation
$-2\ln(\mathcal{L})\varpropto \chi^2$, where $\chi$ is the chi-square function given by
\begin{equation}
\chi^2_{\rm uncorr} =
\sum_{i}^{N_{dat}}\left(\frac{\mathcal{D}-\mathcal{M}}{\sigma_{\mathcal{D}}}
\right)^2\,,
\end{equation}
for the samples OHD, HIIG, SLS because the measurements are considered to be uncorrelated. $N_{dat}$ is the number of points of dataset $\mathcal{D}$, $\sigma_{\mathcal{D}}$ is the estimated uncertainty for each dataset, and $\mathcal{M}$ represents the theoretical quantity of that observable \textbf{based on $E_4$ presented in \eqref{E4} and \eqref{eqN340} for $\Lambda\neq 0$ and $\Lambda=0$ models respectively.} As SNIa and BAO datasets contain correlated points, the figure of merit is built as
\begin{equation}
\chi^{2}_{\rm corr} = \Delta\vec{x}\cdot{\rm C}^{-1}\cdot \Delta\vec{x}^{T}\,,
\end{equation}
where $\Delta\vec{x}$ is the difference between the observational and theoretical quantities, and $\rm C^{-1}$ is the covariance matrix. It is worth to mention that a nuisance parameter is presented in the SNIa data and it is convenient to marginalize over it to reduce the uncertainties. Thus, the figure of merit for SNIa data is
\begin{equation}
\chi^2_{\rm SNIa} = a + \ln\left( \frac{e}{2\pi}\right) - \frac{b^2}{e}\,,
\end{equation}
where $a$, $b$, and $e$ are functions of $\Delta\vec{x}$ and $\rm C^{-1}$. For
more details on these expressions see \citet{Motta:2021hvl}.
Finally, we perform a joint analysis through the sum of the function-of-merits
of each data sample, namely
\begin{equation}
\chi^2_{\rm Joint}=\chi^2_{\rm OHD}+\chi^2_{\rm SLS}+\chi^2_{\rm HIIG}+\chi^2_{\rm SNIa}+\chi^2_{\rm BAO},
\end{equation}
where subscripts indicate the dataset under consideration.
\subsection{Results }
Performing the full confrontation of the scenario we construct the
corresponding log-likelihood contours at $68\%$ ($1\sigma$) and
$99.7\%$ ($3\sigma$) confidence level (CL), and we present them in
Fig. \ref{fig:contours} alongside the 1D posterior
distribution.
Moreover, in Table \ref{tab:bestfits} we
show the mean values and the
uncertainties at $1\sigma$ confidence level for the parameters $h$,
$\Omega_m^{(0)}$ and $\beta$ for both $\Lambda \neq 0$
and $\Lambda=0$ cases.
\begin{figure*}
\centering
\includegraphics[width=0.6\textwidth]{contour1_kadasiano_cc1.pdf}\\
\includegraphics[width=0.6\textwidth]{contour1_kadasiano_cc0.pdf}
\caption{
Two-dimensional log-likelihood contours at 68\% and 99.7\% confidence level
(CL), alongside the corresponding 1D posterior distribution of the free
parameters, in Kaniadakis horizon entropy cosmology, for $\Lambda\neq0$
(upper panel) and $\Lambda=0$
(lower panel). We use the various
datasets described in the text, as well as the joint analysis.
}
\label{fig:contours}
\end{figure*}
\begin{table*}
\centering
\begin{tabular}{|lccccccc|}
\hline
Sample & $\chi^2_{\rm min}$ & $h$ & $\Omega_m^{(0)}$ & $\beta$ & $\Delta$AICc
& $\Delta$BIC & $\Delta$DIC \\
\hline
\multicolumn{8}{|c|}{Case $\Lambda \neq 0$} \\ [0.9ex]
OHD & $19.25$ & $0.699^{+0.033}_{-0.034}$ & $0.354^{+0.072}_{-0.061}$ &
$-0.004^{+1.259}_{-1.255}$ & $7.6$ & $8.2$ & $-4.5$\\ [0.9ex]
BAO & $2.91$ & $0.599^{+0.272}_{-0.270}$ & $0.302^{+0.027}_{-0.023}$ &
$-0.016^{+1.596}_{-1.594}$ & $14.1$ & $1.9$ & $0.2$\\ [0.9ex]
SLS & $216.52$ & $0.608^{+0.268}_{-0.277}$ & $0.077^{+0.064}_{-0.042}$ &
$-0.006^{+2.550}_{-2.526}$ & $5.5$ & $8.3$ & $-5.2$ \\ [0.9ex]
HIIG & $452.96$ & $0.722^{+0.018}_{-0.018}$ & $0.408^{+0.151}_{-0.137}$ &
$0.043^{+2.510}_{-2.562}$ & $19.3$ & $22.3$ & $-19.6$\\ [0.9ex]
SNIa & $1042.99$ & $0.598^{+0.273}_{-0.270}$ & $0.359^{+0.126}_{-0.055}$ &
$0.009^{+1.108}_{-1.124}$ & $9.0$ & $14.0$ & $-14.6$\\ [0.9ex]
Joint & $1743.48$ & $0.708^{+0.012}_{-0.011}$ & $0.283^{+0.016}_{-0.015}$ &
$-0.011^{+0.517}_{-0.507}$ & $2.9$ & $8.1$ & $0.5$ \\ [0.9ex]
\hline
\multicolumn{8}{|c|}{Case $\Lambda = 0$} \\
OHD & $14.56$ & $0.701^{+0.029}_{-0.030}$ & $0.353^{+0.057}_{-0.050}$ &
$1.138^{+0.043}_{-0.051}$ & $0.5$ & $0.0$ & $0.0 $ \\ [0.9ex]
BAO & $2.33$ & $0.602^{+0.272}_{-0.273}$ & $0.297^{+0.023}_{-0.021}$ &
$1.186^{+0.017}_{-0.020}$ & $3.6$ & $-0.4$ & $-0.3 $ \\ [0.9ex]
SLS & $212.86$ & $0.596^{+0.276}_{-0.270}$ & $0.057^{+0.031}_{-0.027}$ &
$1.373^{+0.020}_{-0.023}$ & $-0.2$ & $-0.3$ & $0.0 $\\ [0.9ex]
HIIG & $435.64$ & $0.721^{+0.018}_{-0.018}$ & $0.298^{+0.050}_{-0.045}$ &
$1.185^{+0.038}_{-0.043}$ & $-0.1$ & $-0.2$ & $-0.1 $\\ [0.9ex]
SNIa & $1036.48$ & $0.596^{+0.275}_{-0.271}$ & $0.402^{+0.023}_{-0.023}$ &
$1.093^{+0.021}_{-0.021}$ & $0.5$ & $0.5$ & $0.5 $ \\ [0.9ex]
Joint & $1753.03$ & $0.715^{+0.012}_{-0.012}$ & $0.326^{+0.015}_{-0.015}$ &
$1.161^{+0.013}_{-0.013}$ & $10.4$ & $10.4$ & $10.4 $\\ [0.9ex]
\hline
\end{tabular}
\caption{Best-fit values and their $68\%$ CL uncertainties for Kaniadakis
horizon entropy cosmology with $\Lambda\neq0$ (upper panel) and $\Lambda=0$
(lower panel) employing the data sets: OHD ($31$ data points), BAO (6 data points), SLS (143 data points), HIIG (181 data points), SNIa (1048 data points) and the joint analysis of them. }
\label{tab:bestfits}
\end{table*}
\begin{figure*}
\centering
\includegraphics[width=0.31\textwidth]{plot_Hz_kan_cc1_joint.pdf}
\includegraphics[width=0.31\textwidth]{plot_qz_kan_cc1_joint.pdf}
\includegraphics[width=0.31\textwidth]{plot_jz_kan_cc1_joint.pdf}\\
\includegraphics[width=0.31\textwidth]{plot_Hz_kan_cc0_joint.pdf}
\includegraphics[width=0.31\textwidth]{plot_qz_kan_cc0_joint.pdf}
\includegraphics[width=0.31\textwidth]{plot_jz_kan_cc0_joint.pdf}\\
\caption{Upper panel, left to right: reconstruction of the $H(z)$,
$q(z)$,
and $j(z)$,
in Kaniadakis horizon entropy cosmology with $\Lambda
\neq 0$. Lower panel:
same as before for the case $\Lambda = 0$. We have used the bound obtained from the joint
analysis, and the shaded regions denote the uncertainties at $1\sigma$.
For completeness, the red square-points
represent the results of $\Lambda$CDM cosmology with $h=0.6766$ and
$\Omega_m^{(0)}=0.3111$
\citep{Planck:2018vyg}.}
\label{fig:Hz_and_qz}
\end{figure*}
As
can be seen, the
bounds estimated from each data sample are consistent among
themselves,
although the SLS dataset provides lower values for $\Omega_m^{(0)}$. The joint constraints
$h=0.708^{+0.012}_{-0.011}$ ($h=0.715^{+0.012}_{-0.012}$) for the case $\Lambda
\neq 0$ ($\Lambda=0$), are consistent at $2.67\sigma$ ($3\sigma$) with the one
estimated from the CMB anisotropies \citep{Planck:2018vyg} and at $1.74\sigma$
($1.36\sigma$) with the one from SH0ES \citep{Riess:2019}.
Hence, the scenario of Kaniadakis horizon entropy cosmology can offer an
alleviation to the $H_0$ tension, providing a value in between its local
measurements, and it indirect estimation from the early stages of the Universe.
Concerning Kaniadakis parameter, we
find that, when $\Lambda \neq
0$, the combination of the data samples constrains
$\beta=-0.011^{+0.515}_{-0.507}$, namely $\beta$ is constrained
around 0 as expected,
the value in which Kaniadakis entropy
becomes the standard Bekenstein-Hawking one.
However, when $\Lambda=0$, the joint constraint yields $\beta=1.161^{+0.013}_{-0.013}$,
which is expected, as we mentioned above,
because in the absence of an explicit
cosmological constant one needs a significant deviation from standard
cosmology to describe the Universe acceleration.
Finally, note that due to equation \eqref{corr} that holds in the $\Lambda=0$
case, we acquire a
correlation between $ \Omega_m^{(0)}$ and the Kaniadakis parameter $\beta$ in
the lower panel of Fig. \ref{fig:contours}.
Let us make a comment on the predicted entropy today, since this is
possible to be calculated through Eq. \eqref{kentropy}. According to our
model, and imposing for the horizon area of our Universe its present value, we
arrive at the value $S_K\sim1.44\times10^{99}$ m$^2$Kg s$^{-2}$K$^{-1}$
(for $\Lambda\neq0$) and $S_K\sim3.15\times10^{99}$m$^2$Kg s$^{-2}$K$^{-1}$
(for $\Lambda=0$). In comparison, for the standard Bekenstein-Hawking
entropy, we have $S_{BH}\sim2.83\times10^{99}$ m$^2$Kg s$^{-2}$K$^{-1}$
($\Lambda\neq0$) and $S_{BH}\sim2.79\times10^{99}$m$^2$Kg s$^{-2}$K$^{-1}$
($\Lambda=0$). Therefore, the corresponding ratio is $S_K/S_{BH}\simeq0.5$ for
$\Lambda\neq0$ and $S_K/S_{BH}\simeq1.12$ for $\Lambda=0$, which implies a
small difference between Kaniadakis and Bekenstein-Hawking entropies.
Due to the competitive qualities of the fits obtained from both scenarios, it
would be interesting to statistically compare them
with the concordance
$\Lambda$CDM cosmology. In order to achieve this, we apply the standard
criteria, namely the Akaike information criterion corrected for small samples
\citep[AICc,][]{AIC:1974,
Sugiura:1978, AICc:1989} and
the Bayesian information criterion
\citep[BIC,][]{schwarz1978}, since
$\Lambda\neq 0$ model contains one extra free parameter over $\Lambda$CDM. The
AICc and BIC are defined as ${\rm AICc}= \chi^2_{\text{min}}+2k +(2k^2+2k)/(N-k-1)$ and
${\rm BIC}=\chi^2_{\text{min}}+k\ln(N)$ respectively, where $\chi^2_{\text{min}}$ is the
minimum of the $\chi^2$ function, $N$ is the size of the dataset and $k$ is the
number of free parameters. Following the rules described in
\citet{hernandezalmada2021kaniadakis}, we find that $\Lambda=0$ model and
$\Lambda$CDM are statistically equivalent based on AICc ($\Delta$AICc$<4$), when the sample are
treated separately, but show a strong evidence against ($6<{\rm BIC}<10$) the scenario when the
joint analysis is applied. On the other hand, although AICc suggests that
$\Lambda \neq 0$ model and $\Lambda$CDM are statistically equivalent in the
joint analysis, BIC indicates that there is a strong evidence against
the candidate model.
Additionally, for the two models we find
that the $\Lambda=0$ case is preferred by separate datasets, while the
$\Lambda\neq 0$ case is statistically preferred for the combined data analysis.
For completeness, we additionally calculate
the Deviance information
criterion \citep[DIC,][]{Spiegelhalter:2002, Kunz:2006, Liddle:2007}. This is
defined as ${\rm DIC} = D(\bar{\theta}) + 2p_D$,
where $D(\bar{\theta}) = \chi^2(\bar{\theta})$ is the Bayesian deviation,
$p_D = \bar{D}(\theta) - D(\bar{\theta})$ is the Bayesian complexity, which
represents the number of effective degrees of freedom, and $\bar{\theta}$ is the
mean value of the parameters.
The advantage of DIC is its use of the full
log-likelihood sample instead of only the maximum log-likelihood (or minimum
$\chi^2$) as AICc and BIC do. Based on the Jeffreys scale \citep{Jeffreys:1961},
for $\Delta{\rm DIC}<2$ both models are statistical equivalent. In contrast,
$2<\Delta {\rm DIC}<6$ suggests a moderate tension between models, being the one
with lower value of DIC the best one, and $\Delta {\rm DIC}> 10$ implies a
strong tension between the two models. We find that the $\Lambda \neq 0$ case
and $\Lambda$CDM scenario are statistical equivalent for BAO, they have a
moderate tension for OHD and SLS, and a strong tension for HIIG and SNIa. On the
other hand, the $\Lambda \neq 0$ case and $\Lambda$CDM are statistical
equivalent for OHD, BAO, HIIG, and SNIa. In summary, we confirm the results
obtained for the Joint analysis by AICc and BIC for both $\Lambda\neq 0$ and
$\Lambda=0$ models. It is worth to mention that when a posterior distribution
presents a bimodal shape or is asymmetric for a parameter, $p_D$ yields negative
values and thus DIC may not be a good criterion. This situation is mainly
presented for $\beta$ in the $\Lambda \neq 0$ case in separate datasets.
As a next step, we use the constraints from the joint analysis to reconstruct
the three cosmographic parameters, namely the Hubble, $H(z)$,
the
deceleration, $q(z)$, and jerk, $j(z)$, parameters according
to (\ref{q}), (\ref{j}). The cosmic
evolution of parameters is shown in Fig.
\ref{fig:Hz_and_qz}. Thus, we report the current values of
$q_0=-0.610^{+0.028}_{-0.035}$ ($-0.708^{+0.016}_{-0.016}$) for the
deceleration parameter, and $j_0 = 1.041^{+0.051}_{-0.036}$
($1.137^{+0.014}_{-0.013}$) for the jerk parameter
for the $\Lambda \neq 0$
($\Lambda=0$) scenario.
Furthermore, the transition
redshift
between the deceleration
and the acceleration stages is estimated to be
$z_T=0.715^{+0.042}_{-0.041}$ ($0.652^{+0.032}_{-0.031}$), which is in agreement
with the one obtained by $\Lambda$CDM as shown in Fig. \ref{fig:Hz_and_qz}.
Note that the jerk
parameter evolution reveals the dynamical equation of
state of the effective dark energy.
Finally,
to investigate in more detail the Hubble tension, we apply a
new diagnostic, called $\mathbf{\mathbb{H}}0(z)$ diagnostic, defined by
\citep{H0diagnostic:2021}
\begin{equation}
\mathbf{\mathbb{H}}0(z) = \frac{H(z)}{E_{\rm \Lambda CDM}(z)}\,,
\end{equation}
where $H(z)$ is the Hubble function evolution in a given cosmological scenario
alternative to $\Lambda$CDM,
and $E_{\rm \Lambda CDM}(z)$ is the
dimensionless Hubble parameter of $\Lambda$CDM paradigm. This diagnostic
measures a possible deviation of $H_0$ from its $\Lambda$CDM value. Concerning
a flat $\Lambda$CDM,
a non-constant path of $\mathbf{\mathbb{H}}0(z)$
within error bars suggests a modification of the Planck-$\Lambda$CDM
scenario. In Fig. \ref{fig:H0diagnostic} we depict the obtained
results.
As we observe, there is an agreement within $1\sigma$ between flat-$\Lambda$CDM cosmology and Kaniadakis cosmology for $z\gtrsim 0.7$ in the $\Lambda\neq 0$ case, and for $0.7 \lesssim z \lesssim 1.3$ in the $\Lambda= 0$ case.
Additionally, it is interesting that the current value
$\mathbf{\mathbb{H}}0(z=0)$ for both models are consistent with the one obtained
by SH0ES \citep{Riess:2019}, and that the $\Lambda\neq0$ model has a trend to the Planck
value in the past \citep{Planck:2018vyg}. This is another verification that the
scenario of Kaniadakis horizon entropy cosmology may offer an alleviation to
the $H_0$ tension.
Nevertheless, to further investigate whether both Kaniadakis models can
alleviate the Hubble tension, a parameter estimation using the linear
perturbation equations together with CMB data should be performed.
We close this section by investigating one important process in every
cosmological scenario: the Big Bang Nucleosynthesis (BBN), since
the production of light elements in
the early Universe can be
affected in non-standard cosmologies \citep{Pospelov:2010ARNPS,
Barrow:2021PhLB}. Considering that the freeze-out of the light elements occurs
when the weak interaction rates are lower than $H(z)$, a simple test to
guarantee that the BBN is not spoiled is to require that the deviation $\delta
H(z)$ with respect to the standard Hubble expansion rate at the BBN epoch should
be small. Although in the Friedmann equations mentioned above we have not
included a radiation component, this can be added and we can perform the
analysis by expanding $E_4$ around $\beta=0$ in Eq.
\eqref{E4} (resp. Eq. \eqref{eqN340}) and neglecting the fourth order error
terms, resulting to
\begin{small}
\begin{align}
& E(z)= \left\{
\begin{array}{c}
\underbrace{ \sqrt{\Omega_m^{(0)} (z+1)^{{3 }}+
\Omega_\Lambda^{(0)}}}_{\text{value from}\; \Lambda {\rm CDM}}+ \underbrace{\frac{\beta ^2}{4}
\left[\Omega_m^{(0)} (z+1)^{{3 }}+
\Omega_\Lambda^{(0)}\right]^{-3/2}}_{\text{correction}}, \; \Lambda \neq 0 \\
\underbrace{ \sqrt{\Omega_m^{(0)} } (z+1)^{{3/2}}}_{\text{value from}\; \Lambda {\rm CDM}}+ \underbrace{\frac{\beta ^2}{4}
\left[\Omega_m^{(0)}\right]^{-3/2} (z+1)^{{-1/2}}}_{\text{correction}}, \; \Lambda=0
\end{array}\right..
\label{EQ_53}
\end{align}
\end{small}
Hence, we can study both Kaniadakis models ($\Lambda \neq 0$ and $\Lambda=0$)
at $z\sim10^{10}$ (approximately BBN era).
We find that for $\Lambda\neq 0$, the model is consistent with the Big Bang
Nucleosynthesis (BBN) constraints, since the correction term at
$z\sim10^{10}$ is of the order of $\sim 10^{-49}$, dominating the standard
cosmology and not producing significant effects in the formation of light
elements.
In the case $\Lambda=0$, the correction is larger, and calculations at
$z\sim10^{10}$ are of the order $\sim10^{-5}$. However, such corrections are
still subdominant, allowing the production of light elements. A further
analysis could be performed following
\citet{Capozziello:2017bxm,Barrow:2021PhLB,Asimakis:2021yct}.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{plot_H0diag_kan_cc1_joint.pdf}
\includegraphics[width=0.45\textwidth]{plot_H0diag_kan_cc0_joint.pdf}
\caption{The $\mathbf{\mathbb{H}}0(z)$ diagnostic for Kaniadakis horizon
entropy cosmology with
$\Lambda \neq 0$ (upper panel) and $\Lambda = 0$ (lower panel). We have used
the bound obtained from the joint
analysis, and the shaded regions denote the uncertainties at $1\sigma$.
For completeness, the red square-points
represent the results of $\Lambda$CDM cosmology with $h=0.6766$ and
$\Omega_m^{(0)}=0.3111$
\citep{Planck:2018vyg}.}
\label{fig:H0diagnostic}
\end{figure}
\section{Dynamical system and stability analysis} \label{sec:SA}
In this section we perform a full dynamical system analysis in order to
investigate the global dynamics of cosmological scenarios, and obtain
information on the Universe evolution independently of the initial
conditions.
In the dynamical system formulation, one starts from local analysis of the differential equation $\mathbf{x}'(\tau)={\bf
X}(\mathbf{x})$, where $\mathbf{x}$ is the state vector, and $\tau$ a convenient time variable, near an equilibrium point $\mathbf{x}=\bar{\mathbf{x}}$, and
progressively extends the investigated regions of the phase and of the
parameter space. Assuming that the vector field ${\bf
X}(\mathbf{x})$ has continuous partial
derivatives, the process of determining the local behavior is based on the
linear approximation of the vector field ${{\bf X}(\mathbf{x}) \approx {\bf
DX}(\bar{\mathbf{x}})(\mathbf{x}-\bar{\mathbf{x}})}$ where ${\bf DX}(\bar{\mathbf{x}})$ is the Jacobian of the vector field at the
equilibrium point $\bar{\mathbf{x}}$, which is referred to as the {\it
linearization of the dynamical equations at the equilibrium point}. In this neighborhood we acquire the system $\mathbf{x}'(\tau)={ {\bf
DX}(\bar{\mathbf{x}})(\mathbf{x}-\bar{\mathbf{x}})}$. Each of
the equilibrium points can be classified according to the real parts of the eigenvalues of ${\bf
DX}(\bar{\mathbf{x}})$ (if none of these are zero). Thus, this approach provides a general description of
the phase space of all possible solutions of the system, their equilibrium
points and stability, as well as the asymptotic solutions
\citep{Ellis,Ferreira:1997au,Copeland:1997et,Perko,Coley:2003mj,Copeland:2006wr,
Chen:2008ft,Cotsakis:2013zha,Giambo:2009byn,Papagiannopoulos:2022ohv}. If some
real parts of the eigenvalues are zero, the equilibrium point is nonhyperbolic,
and the analysis through linearization fails. Then, we use numerical tools for
the analysis.
In the following subsections we perform the global dynamical system analysis for the two cases, namely $\Lambda \neq 0$ and $\Lambda = 0$.
\subsection{Case I: $\Lambda \neq 0$}
Defining the dimensionless variables $\theta, T$ as
\begin{align}
\theta= \arctan \left( 1- \frac{8 \pi G \rho_{DE}}{3 H^2 }\right), \; \theta\in \left[-\frac{\pi}{2},\frac{\pi}{2}\right],\quad T= \frac{H_0}{H+ H_0},
\label{Tthetavar}
\end{align}
with
\begin{equation}
\Omega_m:= \frac{8 \pi G \rho_m}{3 H^2 } = \tan \left( \theta \right),
\end{equation}
then equation \eqref{gfe2} becomes
\begin{small}
\begin{align}
\frac{\Lambda}{3 H_0^2 }= \frac{(1-T)^2 \left\{\cosh \left[\frac{T^2 \beta
}{(1-T)^2}\right]-\tan (\theta )\right\}}{T^2}-\beta \text{shi}\left[\frac{T^2
\beta
}{(1-T)^2}\right].
\end{align}
\end{small}
It proves convenient to
introduce the new time derivative as
\begin{align}
f^{\prime} \equiv \frac{d f}{d \tau}= \frac{\cosh \left(\frac{\pi K}{G
H^2}\right)}{H} \dot{f} .
\end{align}
Therefore, we finally extract the dynamical system
\begin{align}
& {\theta}^{\prime}(\tau)= 3 \sin (\theta ) \left\{\sin (\theta )-\cos (\theta
) \cosh \left[\frac{T^2 \beta
}{(1-T)^2}\right]\right\}, \label{eq4.10}\\
& T^{\prime}(\tau)= \frac{3}{2} (1-T) T \tan (\theta). \label{eq4.9}
\end{align}
Note that this system diverges at $T=1$ and at $\theta=\pm\pi/2$.
Lastly, the deceleration parameter (\ref{q}) is written as
\begin{equation}
q:= -1-\frac{\dot{H}}{H^2}= -1 + \frac{3}{2} \tan (\theta )
\text{sech}\left[\frac{\beta T^2}{(1-T)^2}\right].
\end{equation}
Note that, for an expanding universe ($H>0$), we have that $T\in[0,1]$, while
$\theta$
is a periodic coordinate with period $\pi$, and thus we can set
$\theta\in\left[-{\pi}/{2}, {\pi}/{2}\right]$ (modulo a periodic shift $c \pi,
\; c\in\mathbb{Z}$). Moreover, the physical condition $0\leq \Omega_m\leq1$
implies that the region of physical interest is $\theta \in[0, \pi/4]$ (modulo
a periodic shift $c \pi, \; c\in\mathbb{Z}$). The non-physical region
$\Omega_m>1$ is $\theta\in(\pi/4, \pi/2]$ (modulo a periodic shift $c \pi,
\; c\in\mathbb{Z}$).
Hence, we have obtained a global phase-space formulation.
For the representation of the flow of \eqref{eq4.10} and \eqref{eq4.9}, we integrate in the variables $T, \theta$ and project in a compact set
using the ``cylinder-adapted'' coordinates
\begin{equation}
\label{cylinder}
\mathbf{S}: \begin{cases}
x = \cos ( \theta),\\
y = \sin ( \theta),\\
z = T,
\end{cases}
\end{equation}
with $0\leq T\leq 1, \theta \in [-\pi, \pi]$,
with inverse
$
\theta = \arctan \left({y}/{x}\right)$, and $ T =z$.
Thus, the region of physical interest is $\theta \in [0, \pi/4]$,
modulo
a periodic shift $c \pi, \; c\in\mathbb{Z}$.
\begin{table}
\centering
\begin{tabular}{|c|c|c|c|c|}\hline
Label & $\theta$ & $T$& Existence & Stability \\\hline
$dS_{+}$ & $ 2 \pi c_1$ & arbitrary & $c_1\in \mathbb{Z}$ & stable \\
$dS_{+}^{(0)}$ & $ 2 \pi c_1 $ & $0$ & $c_1\in \mathbb{Z}$ & stable \\
$dS_{+}^{(1)}$ & $ 2 \pi c_1 $ & $1$ & $c_1\in \mathbb{Z}$ & stable \\
$dS_{-}$ & $\pi(1 + 2 c_1) $ & arbitrary & $c_1\in \mathbb{Z}$ & stable \\
$dS_{-}^{(0)}$ & $\pi(1 + 2 c_1) $ & $0$ & $c_1\in \mathbb{Z}$ & stable \\
$dS_{-}^{(1)}$ & $\pi(1 + 2 c_1) $ & $1$ & $c_1\in \mathbb{Z}$ & stable \\
$M_{-}^{(0)}$ & $2 \pi c_1-\frac{3 \pi }{4} $ & $0$ & $c_1\in \mathbb{Z}$ & unstable \\
$M_{+}^{(0)}$& $2 \pi c_1+\frac{\pi }{4} $ & $0$ & $c_1\in \mathbb{Z}$ & unstable \\
$M_{-}^{(1)}$ & $2 \pi c_1-\frac{3\pi }{4} $ & $1$ & $c_1\in \mathbb{Z}, \beta=0$ & saddle \\
$M_{+}^{(1)}$ & $2 \pi c_1+\frac{\pi }{4} $ & $1$ & $c_1\in \mathbb{Z}, \beta=0$ & saddle \\\hline
\end{tabular}
\caption{The equilibrium points of the dynamical system \eqref{eq4.10}
and \eqref{eq4.9} of Kaniadakis horizon entropy cosmology with $\Lambda\neq0$.
We use $dS$ to denote the de Sitter, dark-energy dominated solutions, and $M$ to denote the matter-dominated ones.
}
\label{Eigen-1}
\end{table}
We proceed by extracting the equilibrium points and
characterizing their stability. There are two equivalent hyperbolic
equilibrium points $M_{\pm}$ for which $q=1/2$,
i.e. they are
associated with matter domination, and two equilibrium points
$dS_{\pm}$ corresponding to dark-energy dominated de-Sitter solutions for which
$q = -1$. The equilibrium points of the dynamical system \eqref{eq4.10}
and \eqref{eq4.9} of Kaniadakis horizon entropy cosmology with $\Lambda\neq0$ are presented in Table \ref{Eigen-1}.
\begin{figure}
\centering
\includegraphics[scale=0.6]{PHASE-PLOT2.pdf}
\caption{Phase-space plot of the dynamical system \eqref{eq4.10}
and \eqref{eq4.9} of Kaniadakis horizon entropy cosmology with $\Lambda\neq0$,
for the best-fit value of Kaniadakis parameter obtained by the observational
analysis, namely for $\beta=-0.011$. Upper panel:
unwrapped solution space. Lower panel: projection over the
cylinder $\mathbf{S}$ defined in Cartesian coordinates $(x,y,z)$ through
\eqref{cylinder}. At late times the Universe results
in a dark-energy dominated, de Sitter solution, while
the past attractor is the matter-dominated epoch.}
\label{fig:my_label2}
\end{figure}
In Fig. \ref{fig:my_label2} we display an unwrapped solution space of the system \eqref{eq4.10} and \eqref{eq4.9} (upper
panel), and the projection over the cylinder $\mathbf{S}$, defined in Cartesian coordinates $(x,y,z)$ by \eqref{cylinder}, for the best fit value $\beta=-0.011$ obtained through the observational analysis.
For the points that are non-hyperbolic, their stability is analyzed
numerically. The two dashed lines, indicated by $dS_{-}$ (blue) and
$dS_{+}$ (red), are the late-time de Sitter attractors. The early-time attractors
are $M_{\pm}^{(0)}$ for which $q=1/2$,
and they correspond to
matter-dominated solutions. Hence,
at late times the Universe results
in a dark-energy dominated solution, while
the past attractor of the Universe is the matter-dominated epoch. At
the intersection of the invariant set $T=1$ with the singular lines $\theta=\pm
\pi/2$ we obtain the equilibrium points $L_{\pm}$. Considering that equations
\eqref{eq4.10} and \eqref{eq4.9} diverge at $L_{\pm}$, we should
introduce suitable variables for the analysis.
For the analysis at $T=1$ it proves convenient to define the variable
\begin{equation}
\Phi= \left\{1+\exp \left[\frac{|\beta|
T^2}{(1-T)^2}\right]\right\}^{-1}, \Phi \in[0,1],
\end{equation}
as well as the time rescaling
\begin{equation}
f^{\prime} \equiv \frac{d f}{d \zeta}= (1-\Phi)^2 \frac{d f}{d
\eta}= \frac{\tanh ^2\left(\frac{\pi
|K|}{2 G H^2}\right)+1}{4 H} \dot{f}.
\end{equation}
Hence, using also the variable $\theta$ from (\ref{Tthetavar}),
we finally
obtain the autonomous system
\begin{align} & \theta^{\prime}(\zeta)= -\frac{3}{2} \sin (\theta ) \left[2 (\Phi -1) \Phi
(\sin (\theta )+\cos (\theta ))+\cos (\theta )\right], \label{4.39}\\
& \Phi^{\prime}(\zeta)= 3 (1-\Phi)^2 \Phi ^2 \tan (\theta ) \ln
\left[\Phi/(1-\Phi)\right]. \label{4.38}
\end{align}
\begin{figure}
\centering
\includegraphics[scale=0.9]{T1.pdf}
\caption{Phase-space plot of the system \eqref{4.39}-\eqref{4.38} of
Kaniadakis horizon entropy cosmology with $\Lambda\neq0$, for dust matter. The
late attractor corresponds to $ \theta=0$, and thus to a dark-energy
domimated solution with
$\Omega_{DE}= 1$. }
\label{T1}
\end{figure}
In Fig. \ref{T1} we depict the phase-space flow of the system
\eqref{4.39}-\eqref{4.38}. Asymptotically, $\theta\rightarrow 0$ and $\Phi$ tends to a constant $\Phi_0$.
Therefore, the late attractor corresponds to the dark-energy dominated solution
with $\Omega_{DE}= 1$. The current values $ \Phi_{0}= 1/(e^{\left| \beta \right| }+1)$, $\theta_0=\text{arctan}\left(\Omega_m^{(0)}\right)$ leads to the de Sitter solution $a(t)=e^{H_0 (t-t_U)}$.
\subsubsection{Heteroclinic sequences}
\label{section4.1.1}
In the phase portrait of a dynamical system, a heteroclinic orbit is a
path in phase space that joins two different equilibrium points. If the
equilibrium points at the start and end of the orbit are the same, the orbit is
a homoclinic orbit \citep{Guckenheimer}.
From the above analysis we can see that the invariant sets $T=0$ and
$T=1$ are of interest in the determination of possible heteroclinic sequences.
The direction of the flow can be determined by considering the monotonic
function
\begin{equation}
M_1= \frac{T}{1-T},\quad M_1'(\tau)=\frac{3 \tan (\theta )}{2} M_1.
\end{equation}
In if $\tan(\theta)<0$, the orbits move from $T=1$ to $T=0$, and if
$\tan(\theta)>0$, the orbits move from $T=0$ to $T=1$. In the invariant manifold
$T=0$ ($H\rightarrow \infty$) the dynamics is given by the one-dimensional
flow
\begin{equation}
\label{one-FIG4A}
{\theta}^{\prime}(\tau)= 3 \sin (\theta ) \left(\sin (\theta )-\cos (\theta
) \right).
\end{equation}
In Fig. \ref{FIG4A} we present the one-dimensional dynamical system
\eqref{one-FIG4A}, in which we can see the heteroclinic sequences $M_+^{(0)}
\to d S_{+}^{(0)}$
and $M_-^{(0)} \to d S_{-}^{(0)}$.
\begin{figure}
\centering
\includegraphics[scale=0.3]{FIG4A.pdf}
\caption{Phase-space diagram of the one-dimensional dynamical system \eqref{one-FIG4A} of
Kaniadakis horizon entropy cosmology with $\Lambda\neq 0$, for dust matter and any value $\beta$. }
\label{FIG4A}
\end{figure}
Similarly, analyzing the one-dimensional flow in the invariant
set $T=1$, that corresponds to $\Phi=0$, we find that the dynamics on this
invariant set is given by the one-dimensional dynamical system
\begin{equation}
\label{one-FIG4B}
\theta^{\prime}(\zeta)= -\frac{3}{2} \sin (\theta ) \cos (\theta),
\end{equation}
\begin{figure}
\centering
\includegraphics[scale=0.3]{FIG4B.pdf}
\caption{Phase-space diagram of the one-dimensional dynamical system \eqref{one-FIG4B} of
Kaniadakis horizon entropy cosmology with $\Lambda\neq 0$, for dust matter and any value $\beta$. }
\label{FIG4B}
\end{figure}
which has a behavior shown in Fig. \ref{FIG4B}, where the
heteroclinic sequences $L_+ \to d S_{+}^{(1)}$ and $L_- \to d S_{-}^{(1)}$ are presented.
Finally, to find heteroclinic sequences $M_\pm^{(0)} \to d
S_{\pm}^{(1)}$, the intersection of the unstable manifold of $M_\pm^{(0)}$ with
the stable manifold of $d S_{\pm}^{(1)}$ should be analyzed. Since the former
is $\mathbb{R}^2$, then it is required to examine the stable manifold of $d
S_{\pm}^{(1)}$.
This is given locally by the graph
\begin{equation}
\left\{(\Phi, \theta)\in \mathbb{R}^2: \Phi= h(\theta), h(0)=0, h'(0)=0
\right\}, |\theta|<\delta,
\end{equation}
for $\delta>0$ suitably small.
By the invariance of the stable manifold we obtain the quasilinear
differential equation for $h$ given by
\begin{align}
& \frac{3}{2} \sin (\theta ) h'(\theta ) (2 (h(\theta )-1) h(\theta ) (\sin
(\theta )+\cos (\theta ))+\cos (\theta )) \nonumber \\
& +3
(h(\theta )-1)^2 h(\theta )^2 \tan (\theta ) \ln \left(\frac{h(\theta
)}{1-h(\theta )}\right)=0.
\end{align}
Introducing the ansatz
$
h(\theta)= a_1 \theta^2 + a_2 \theta^3 + a_3 \theta^4 + \ldots,
$
we obtain $a_i=0$ at any order. Therefore, the dynamics at the stable manifold
of $d S_{+}^{(1)}$ is given by equation \eqref{one-FIG4B}. Then, it is easy
to construct heteroclinic sequences $M_+^{(0)} \to d S_{+}^{(1)}$ which pass
near the singularity $L_+$ by assuming for instance the initial value $(\Phi,
\theta)=(\varepsilon, \pi/4)$, $\varepsilon \approx 0$ and evolving the system
back and forward in $\zeta$. Similar arguments can be used to construct
heteroclinic sequences $M_-^{(0)} \to d S_{-}^{(1)}$, which pass near the
singularity $L_-$, with the initial value $(\Phi, \theta)=(\varepsilon,
-3\pi/4)$, $\varepsilon \approx 0$.
Summarizing, for $0\leq \theta \leq \pi/2$ (the physical region is $0\leq \theta
\leq \pi/4$), there exists the heteroclinic sequences $M_+^{(0)}\;
(\Omega_m\rightarrow 1, H\rightarrow \infty) \to L_+ (\Omega_m\rightarrow
+\infty, H\rightarrow 0) \to d S_{+}^{(1)} (\text{de Sitter},
\Omega_m\rightarrow 0, H\rightarrow 0)$ and $M_+^{(0)} \to d S_{+}^{(0)}
(\text{de Sitter}, \Omega_m\rightarrow 0, H\rightarrow \infty)$, and in the
region $-\pi \leq \theta \leq -\pi/2$ (the physical region is $-\pi\leq \theta
\leq - 3\pi/4$), there exists the heteroclinic sequences $M_-^{(0)}
(\Omega_m\rightarrow 1, H\rightarrow \infty) \to L_- (\Omega_m\rightarrow
-\infty, H\rightarrow 0) \to d S_{-}^{(1)} (\text{de Sitter},
\Omega_m\rightarrow 0, H\rightarrow 0)$ and $M_-^{(0)} \to d S_{-}^{(0)}
(\text{de Sitter}, \Omega_m\rightarrow 0, H\rightarrow \infty)$.
\subsubsection{Bounce and a turnaround}
Another interesting cosmological possibility is the possible
existence of a bounce and a turnaround
\citep{Saridakis:2007cf,Cai:2012ag,Zhu:2021whu}. Let us assume that, for the
state vector $(a, H, R)$, the field equations can be written as
\begin{align}
& \dot{a}= a H, \label{eq_a}\\
& \dot{H}= \frac{1}{6}\left(R - 12 H^2\right),\label{eq71} \\
& \dot{R}= g(a, H, R),\label{eq_g}
\end{align}
such that the function $g(a, H, R)$ satisfies $g(a, H, R)= - g(a, -H, R)$.
Hence, the system \eqref{eq_a}, \eqref{eq71} and \eqref{eq_g} is invariant under
time inversion $t \mapsto -t$ if also $H \mapsto -H$ and $R \mapsto R$, and by
definition $a\geq 0$. Those solutions can be related to symmetric cyclic
solutions with respect to the origin, chosen to correspond to the
possible bounce point $t_{\text{bounce}}=0$. Therefore, if the bounce
exists, the system \eqref{eq_a}, \eqref{eq71} and \eqref{eq_g} is a reversible
system in the sense that it has a reversing symmetry under time inversion.
Let us
consider the simplest case where there is exactly one bouncing and exactly one
turnaround point. Note that both at the bounce and turnaround points we have
$H=0$. In this case, the line connecting these points and corresponding to $H=0$
defines a plane that separates all points on the trajectory in this phase
space to the ones corresponding to either the expanding ($H > 0$) or contracting
($H < 0$) phase. As discussed in \cite{Pavlovic:2020sei}, it is natural that
in cyclic models the value of the Ricci scalar would approach its maximum around
the bounce and since $\dot{H} >0$, from \eqref{eq71} it follows that this
maximum Ricci scalar value is positive, and moreover that $\ddot{H}=0$ at the
bounce.
In summary, at the bounce we have $R= R_{\text{bounce}} > 0$, $H=0$,
and $a=a_{\text{min}}$. The bounce is then followed by a phase in which $\dot{H}
> 0$, $H>0$, $\dot{a}> 0$, and $\dot{R} < 0$. Then the Universe enters the
phase characterized by $\dot{H}< 0$ and approaches the turnaround point which
is determined by $H=0$, $a=a_{\text{max}}$, and $R= R_{\text{turnaround}} < 0$,
where the last condition follows from Eq. \eqref{eq71}.
To obtain $g(a, H, R)$ in Eq. \eqref{eq_g}, we use Eqs. \eqref{gfe1},
\eqref{matterconsv}, \eqref{H1},
\eqref{H2}, and \eqref{PDE}, where for simplicity we focus on the dust case.
Therefore, we obtain
\begin{align}
& R= 6 \dot{H}+12 H^2= -24 \pi G p_{DE}\nonumber\\
& = 3 \Bigg\{-8 \pi G \rho_{m} \left[\text{sech}\left(\frac{\beta
H_0^2}{H^2}\right)-1\right] \nonumber \\
& +3 \beta
H_0^2 \text{shi}\left(\frac{H_0^2 \beta }{H^2}\right)-3 H^2 \left[\cosh
\left(\frac{\beta
H_0^2}{H^2}\right)-1\right]+\Lambda \Bigg\}.
\end{align}
Introducing the dimensionless quantities
$
E= \frac{H}{H_0}$ and $\mathcal{R}= \frac{R}{12 H_0^2}$,
and using $z$ as the independent variable, we extract the general
system for $(a, E, \mathcal{R})$:
\begin{align}
&\frac{da}{dz}= -a^2, \label{eqAa}\\
&\frac{dE}{dz}= -2\left(\mathcal{R} - E^2\right) \frac{a}{E},\label{eqAb} \\
& \frac{d\mathcal{R}}{dz}= -\frac{9 \beta {\Omega_m^{(0)}}^2 \tanh
\left(\frac{\beta}{E^2}\right)
\text{sech}^2\left(\frac{\beta}{E^2}\right)}{4 E^4 a^5}. \label{eqAc}
\end{align}
Finally, in order to examine whether the above requirements are
fulfilled in the present scenario, we use the best fit values $\beta=-0.011$
and $\Omega_m^{(0)}=0.283$ in \eqref{eqAa}, \eqref{eqAb} and \eqref{eqAc}, and
we find
\begin{align}
& \frac{da}{dz}= -a^2, \label{systAa}\\
& \frac{dE}{dz}=- 2\left(\mathcal{R} - E^2\right)\frac{a}{E},
\label{systAb}
\\
& \frac{d\mathcal{R}}{dz}= -\frac{0.0019822 \tanh
\left(\frac{0.011}{E^2}\right)
\text{sech}^2\left(\frac{0.011}{E^2}\right)}{E^4 a^5} \label{systAc}.
\end{align}
In the case of dust matter and $\Lambda\neq 0$. This system cannot satisfy the
above requirements, and hence the present scenario cannot exhibit bounce and
turnaround solutions.
\subsection{Case II: $\Lambda = 0$}
In the case $\Lambda = 0$, eq. \eqref{gfe2} becomes
\begin{equation}
\frac{8 \pi G \rho_{m}}{3 H^2} =- \frac{3 \pi K \text{shi}\left(\frac{K \pi
}{G
H^2}\right)}{3 G H^2 } + \cosh \left(\frac{\pi K}{G
H^2}\right).
\end{equation}
\begin{figure}
\centering
\includegraphics[scale=0.6]{F1D.pdf}
\caption{Phase-space diagram of the one-dimensional dynamical system
\eqref{one-D} of
Kaniadakis horizon entropy cosmology with $\Lambda=0$, for dust matter and for
the best-fit value of Kaniadakis parameter obtained by the observational
analysis, namely for $\beta=1.161$. The physical region is $0\leq T< 1$. The
equilibrium point $T=0$ is unstable,
dominated by dark-energy, and de Sitter equilibrium point
$T=T_c\approx 0.521$ is stable.}
\label{F1D}
\end{figure}
This expression is used as a definition of $\rho_m$.
If $\beta\neq0$, and re-scaling the time derivative $d/d\nu = (1-T)d/d\tau$, we
obtain
\begin{small}
\begin{align}
& T^{\prime}(\nu)=\frac{3}{2} (1-T)^2 T \cosh \left(\frac{\beta
T^2}{(1-T)^2}\right) - \frac{3}{2}\beta T^3 \text{shi}\left(\frac{T^2 \beta
}{(1-T)^2}\right).\label{one-D}
\end{align}
\end{small}
The equilibrium points of \eqref{one-D} are $T=0$, which is unstable, and the
equilibrium point
$T=T_c$, where $T_c$ is a solution of the transcendental equation
\begin{small}
$(1-T_c)^2 \cosh \left( {\beta T_c^2}/{(1-T_c)^2}\right) - \beta T_c^2
\text{shi}\left( {T_c^2 \beta }/{(1-T_c)^2}\right)=0$
\end{small}, $0<T_c<1$, corresponding to de Sitter solution $a(t)\propto e^{H_0
t \left(\frac{1}{T_c}-1\right)}$, is stable. In Fig. \ref{F1D} we depict a
phase-space plot of the one-dimensional dynamical system \eqref{one-D} of
Kaniadakis horizon entropy cosmology with $\Lambda=0$, for dust matter, and the
join value $\beta=1.161$. Note that all orbits originate from the invariant
subset $T=0$, classically related to the initial
singularity with $H \rightarrow \infty$. The late-time attractor is
$T=T_c\approx 0.521$, and it corresponds to de Sitter solution.
Finally,
in order to examine whether the present scenario exhibits
a bounce, we use the best fit values $\beta=1.161$ and $\Omega_m^{(0)}=0.326$
in \eqref{eqAa}, \eqref{eqAb} and \eqref{eqAc} and we find
\begin{align}
& \frac{da}{dz}= -a^2, \label{systBa}\\
& \frac{dE}{dz}= -2\left(\mathcal{R} - E^2\right) \frac{a}{E},
\label{systBb}\\
& \frac{d\mathcal{R}}{dz}= \frac{0.277619 \tanh
\left(\frac{1.161}{E^2}\right)
\text{sech}^2\left(\frac{1.161}{E^2}\right)}{E^4 a^5}, \label{systBc}
\end{align}
in the case of dust matter
and $\Lambda=0$,
we deduce that the system cannot fulfill the
bounce requirements, and therefore it cannot exhibit bounce and
turnaround solutions.
\section{Summary and discussion} \label{sec:Con}
The present work was devoted to explore Kanadiakis horizon entropy cosmology
which arises from the application of the gravity-thermodynamics conjecture
using the Kaniadakis modified entropy. The resulting modified Friedmann
equations contain extra terms that constitute an effective dark energy
sector.
Moreover, we used data from
Observational Hubble Data, Supernova Type Ia, HII galaxies, Strong Lensing
Systems, and Baryon
Acoustic Oscillations observations, and we applied a Bayesian Markov
Chain Monte Carlo analysis in order to construct the likelihood contours
for the model parameters.
Regarding the Kaniadakis parameter $\beta$, we found that it is constrained
around 0, namely around the value in which standard Bekenstein-Hawking is
recovered. Furthermore, the present matter density parameter $\Omega_m^{(0)}$
is
consistent with the expected value from $\Lambda$CDM scenario, having a lower
value for the $\Lambda\neq0$ case and a slightly higher value for the
$\Lambda=0$ case.
However, the interesting result comes from the constraint
on the normalized Hubble parameter $h$. In particular, for $\Lambda \neq 0$
we extracted $h=0.708^{+0.012}_{-0.011}$ while for $\Lambda=0$ we found
$h=0.715^{+0.012}_{-0.012}$. Thus, the obtained value of $H_0$ for $\Lambda
\neq
0$ deviates $2.67\sigma$ from the
Planck value and $1.74\sigma$ from the SH0ES one, while in the
$\Lambda=0$ case the deviation is
$3\sigma$ from the Planck value and $1.36\sigma$ from the SH0ES one.
Additionally, in order to verify this result in an
independent way, we performed the $\mathbf{\mathbb{H}}0(z)$
diagnostic. Hence, our analysis reveals
that Kaniadakis horizon entropy cosmology is an interesting
candidate to alleviate the $H_0$ tension problem. This is one of the main
results of the present work.
We proceeded by investigating the cosmographic parameters, namely the
deceleration and jerk ones, by using the data in order to reconstruct them in
the redshift region $0<z<2.5$. As we showed, the transition from
deceleration to acceleration happens at $z_T=0.715^{+0.042}_{-0.041}$ for the
$\Lambda \neq 0$ case and at $z_T=0.652^{+0.032}_{-0.031}$ for the
$\Lambda =0$ case, in agreement within $1\sigma$ with that found in
\cite{HerreraZamorano:2020rdh} for $\Lambda$CDM cosmology.
Furthermore,
we applied the AICc and BIC information criteria and we found that
although AICc suggests that
$\Lambda \neq 0$ model and $\Lambda$CDM are statistically equivalent in the
joint analysis, BIC indicates that there is a strong evidence against
the candidate model. Lastly, applying the DIC criterion we found that
the $\Lambda \neq 0$ case and
$\Lambda$CDM are statistical equivalent for BAO, they have a moderate tension
for OHD and SLS, and a strong tension for HIIG and SNIa datasets, while the
$\Lambda \neq 0$ case and $\Lambda$CDM are statistical equivalent for
all datasets.
Finally, we performed a detailed dynamical-system analysis, providing a
general description of the phase-space of all possible solutions of the
system, their equilibrium points and stability, as well as the late-time
asymptotic behavior. As we showed, the Universe past attractor is the
matter-dominated epoch, while at late times the Universe results in the
dark-energy-dominated solution, for both $\Lambda=0$, and $\Lambda \neq0$ cases.
Moreover, we showed that the scenario accepts
heteroclinic sequences, but it cannot lead to bounce and turnaround
solutions.
In summary, the scenario of Kaniadakis horizon entropy cosmology exhibits very
interesting phenomenology and is in agreement with observational behavior.
Hence, it can be an interesting candidate for the description of Nature.
\section*{Acknowledgments}
We thank the anonymous referee for thoughtful remarks and suggestions. A.H.A. thanks to the PRODEP project, Mexico for resources
and financial support and thanks also to the support from Luis Aguilar,
Alejandro de Le\'on, Carlos Flores, and Jair Garc\'ia of the Laboratorio
Nacional de Visualizaci\'on Cient\'ifica Avanzada.
G.L. was funded by Agencia Nacional de Investigaci\'on y Desarrollo - ANID for
financial support through the program FONDECYT Iniciaci\'on grant no. 11180126
and by Vicerrectoría de Investigación y Desarrollo Tecnológico at UCN. J.M. acknowledges the support from ANID project Basal AFB-170002 and ANID REDES
190147. M.A.G.-A. acknowledges support from Universidad Iberoamericana that support with the SNI grant, ANID REDES (190147), C\'atedra Marcos Moshinsky and Instituto Avanzado
de Cosmolog\'ia (IAC). V.M. acknowledges support
from Centro de Astrof\'{\i}sica de Valpara\'{i}so and ANID REDES 190147. This
work is partially supported by the Ministry of Education and Science of the Republic of Kazakhstan, Grant AP08856912. A.D. Millano was supported by Agencia Nacional de Investigación y Desarrollo - ANID-Subdirección de Capital Humano/Doctorado
Nacional/año 2020- folio 21200837 and by Vicerrectoría de Investigación y Desarrollo Tecnológico at UCN.
\section*{Data Availability}
The data underlying this article were cited in Section \ref{subsec:data}.
\bibliographystyle{mnras}
|
{
"timestamp": "2022-03-21T01:31:44",
"yymm": "2112",
"arxiv_id": "2112.04615",
"language": "en",
"url": "https://arxiv.org/abs/2112.04615"
}
|
\section{Introduction}
\Ni Parameter estimations are important steps in parametric statistical modeling. Estimators of parameters can be derived from the maximum likelihood approach, the plug-in methods, the moment methods, etc. Of course, by far Maximum Likelihood Estimators (MLE) are preferred because of the statistical meaning of its derivation.
Moments estimators (\textbf{ME})'s and (\textit{MLE})'s may exists without having closed-form expressions. When a \textit{MLE} does no have a closed-from estimator, the \textit{ME} is a backup solution for authors who wish to a clear idea of the estimation and a quick and more controlled ways of computation. Finding \textit{ME}'s is an important step in modeling. However, deriving the related statistical tests is needed for accepting or rejecting hypotheses.\\
\Ni For more that two parameters, it is more practicable to join the individual normal asymptotic laws for each parameters into one chi-square asymptotic laws which is qualified as omnibus following the Jarque-Berra chi-square asymptotic law.\\
\Ni This motivates us to investigate asymptotic laws of \textit{ME}'s estimators of as much as possible of usual and non-usual statistical laws. The found law should be validated by simulation studies before being proposed to potential users.\\
\Ni For a large review of asymptotic estimations and statistical tests, we refer to \cite{vaart_asymp}, \cite{billingsley}, etc. Especially, for methods including functional empirical process, \cite{vaart} is recommended.\\
\Ni However the main tool used here, but not limited to, is the function empirical process (fep) transformed into a instrument tools in \cite{lofep} (see below). let us begin by giving a few words on that tool and next basic notation.\\
\Ni In this paper, we used the \textit{fep} tool to show direct and efficient ways for deriving asymptotic statistical tests for moment estimators for a selected set of four probability laws. Four these laws, all the computations are given in details. Computer codes for simulations are also provided. We could have treated more statistical distributions. However, we wanted this paper to be a model for researchers who need asymptotic statistical tests. Later, we expect to compose a handbook which includes a great number of laws.\\
\Ni The paper is organized as follows. We will close this introductory section by describing the \textit{lofep} tool in Subsection \ref{ss-fep} and, in Section \ref{omnibus} by showing how to derive omnibus chi-square tests from the Gaussian asymptotic theorems for distributions of more than two parameters. In Section \ref{sec2}, we expose the asymptotic laws of the moments estimators of \textit{gamma}, \textit{uniform}, \textit{beta} and \textit{Fisher} distributions. The proofs and the implementation of the \textit{fep} tool on these distributions are stated in Section \ref{sec4}. In Section \ref{sec3}, we proceed to a simulation study on the asymptotic results and show that the omnibus chi-square tests work fine for small samples. The codes used for the simulations are stated in an appendix from page \pageref{scripts_all}. The paper ends with conclusions and perspectives in Section \ref{sec5}.\\
\subsection{A brief reminder of the \textit{fep}} \label{ss-fep} Let $Z_{1}$, $Z_{2}$, ... be a sequence of independent copies of a random variable $Z$ defined on the same probability space with values on some metric space $(S,d)$. Define for each $n\geq 1,$ the functional empirical process by
\begin{equation*}
\mathbb{G}_{n}(f)=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}(f(Z_{i})-\mathbb{E} f(Z_{i})),
\end{equation*}
\Bin where $f$ is a real and measurable function defined on $\mathbb{R}$ such that
\begin{equation}
\mathbb{V}_{Z}(f)=\int \left( f(x)-\mathbb{P}_{Z}(f)\right)
^{2}dP_{Z}(x)<\infty , \label{var}
\end{equation}
\Bin which entails
\begin{equation}
\mathbb{P}_{Z}(\left\vert f\right\vert )=\int \left\vert f(x)\right\vert
dP_{Z}(x)<\infty \text{.} \label{esp}
\end{equation}
\Bin Denote by $\mathcal{F}(S)$ - $\mathcal{F}$ for short -the class of real-valued measurable functions that are defined on S such
that (\ref{var}) holds. The space $\mathcal{F}$ , when endowed with the addition and the external multiplication by real scalars, is a linear space.
Next, it remarkable that $\mathbb{G}_{n}$ is linear on $\mathcal{F}$, that is for $f$ and $g$ in $\mathcal{F}$ and for $(a,b)\in \mathbb{R}{^{2}}$, we have
\begin{equation*}
a\mathbb{G}_{n}(f)+b\mathbb{G}_{n}(g)=\mathbb{G}_{n}(af+bg).
\end{equation*}
\Bin We have this result
\begin{lemma} \label{lemma.tool.1}
\bigskip Given the notation above, then for any finite number of elements $f_{1},...,f_{k}$ of $\mathcal{S},k\geq 1,$ we have
\begin{equation*}
^{t}(\mathbb{G}_{n}(f_{1}),...,\mathbb{G}_{n}(f_{k}))\rightsquigarrow
\mathcal{N}_{k}(0,\Gamma (f_{i},f_{j})_{1\leq i,j\leq k}),
\end{equation*}
\Bin where
\begin{equation*}
\Gamma (f_{i},f_{j})=\int \left( f_{i}-\mathbb{P}_{Z}(f_{i})\right) \left(f_{j}-\mathbb{P}_{Z}(f_{j})\right) d\mathbb{P}_{Z}(x),1\leq i,j\leq k.
\end{equation*}
\end{lemma}
\Bin \textbf{Proof}. It is enough to use the Cram\'{e}r-Wold Criterion (see for example \cite{billingsley}, page 45), that
is to show that for any $a=^{t}(a_{1},...,a_{k})\in \mathbb{R}^{k},$ by denoting $T_{n}=^{t}(\mathbb{G}_{n}(f_{1}),...,\mathbb{G}_{n}(f_{k})),$ we
have $<a,T_{n}>\rightsquigarrow <a,T>$ where $T$ follows the $\mathcal{N
_{k}(0,\Gamma (f_{i},f_{j})_{1\leq i,j\leq k})$\ law and $<\circ ,\circ >$
stands for the usual product scalar in $\mathbb{R}^{k}.$ But, by the standard central limit theorem in $\mathbb{R}$, we have
\begin{equation*}
<a,T_{n}>=\mathbb{G}_{n}\left( \sum\limits_{i=1}^{k}a_{i}f_{i}\right)= \sum\limits_{i=1}^{k} a_{i}\mathbb{G}_{n}\left(f_{i}\right)
\rightsquigarrow N(0,\sigma _{\infty }^{2}),
\end{equation*}
\Bin where, for $g=\sum_{1\leq i\leq k}a_{i}f_{i}$,
\begin{equation*}
\sigma _{\infty }^{2}=\int \left( g(x)-\mathbb{P}_{Z}(g)\right) ^{2} \ d\mathbb{P}_{Z}(x)
\end{equation*}
\Bin and this easily gives
\begin{equation*}
\sigma _{\infty }^{2}=\sum\limits_{1\leq i,j\leq k}a_{i}a_{j}\Gamma
(f_{i},f_{j}),
\end{equation*}
\Bin so that $N(0,\sigma _{\infty }^{2})$ is the law of $<a,T>.$ The proof is finished.
\subsection{Main notations in the \textit{fep}} \label{omnibus} In the context of this paper, we use univariate samples $X$, $X_1$, $X_2$, $\cdots$, $X_n$, $n\geq$, with common cumulative distribution function (\textit{cdf}) $F_X=F$, defined on the same probability space $(\Omega, \mathcal{A}, \mathbb{P})$. We will usually need the cumulants $m_k$ and the centered moments $\mu_k$ defined by
$$
m_k=\mathbb{E}X^k, \ \ \mu_k=\mathbb{E} (X-\mathbb{E}(X))^k, \ \ k\geq 1
$$
\Bin and their plug-in estimators
$$
\overline{X}_{k,n}=\frac{1}{n}\sum_{j=1}^{n} X_{j}^{k}, \ \ \overline{\mu}_{k,n}=\frac{1}{n}\sum_{j=1}^{n} \left(X_{j}-\overline{X}_n\right)^k, \ \ k\geq 1
$$
\Bin with the special case of the empirical mean $\overline{X}_{k,n}=\overline{X}_{n}$. Also the standard variance
$$
S_{n}^2=\frac{1}{n-1}\sum_{j=1}^{n} \left(X_{j}-\overline{X}_n\right)^2
$$
\Bin will be preferred to the plug-in estimator $\overline{\mu}_{2,n}$ of $\sigma^2=\mathbb{V}ar(X)$. We suppose that any moment of order $k\geq 1$ exists whenever it is used.\\
\Bin The moment method in a parametric estimation related to the studied random variable having $\ell\geq 1$ parameters $(\theta_1,\cdots,\theta_\ell)$ and which generates the sample $\{X_1, \cdots, X_n\}$ consisted in simultaneously solving $\ell$ equations, each of these equations $r \in \{1,\cdots,\ell\}$ being the equality between a cumulant or a moment of order $k_r$ and the corresponding plug-in estimator of the same order $h_r$ where all order $k_j$ are pairwise distinct. In general, it is simpler to take equations between the $\ell^{th}$ first cumulants or moments. The solution, whenever exists and statistics of the empirical cumulant or moments,
$$
\hat \theta_n=(\hat \theta_{1,n},\cdots,\hat \theta_{\ell,n})
$$
\Bin is the vector moment estimator (\textit{ME}).\\
\Ni Once the \textit{ME}'s are found, we will need the joint asymptotic law of the vector $\hat \theta_n$. The tool of the \textit{fep} will greatly help in that target. We will go beyond and derive chi-square tests as much as possible.\\
\Ni The rest of the paper is organized as follows...\\
\subsection{Chi-square law derivation} We are going to show how to derive asymptotic chi-square laws from moment estimators for at least two parameters. In each case below, we treat a two-parameter estimation problem. Suppose that the two parameters are denoted by $a$ and $b$ and their moment estimators are denoted by $\hat a_n$ and $\hat b_n$, $n\geq 2$. We will get in each case a first law in the form: as $n\rightarrow +\infty$,
\begin{equation}
\left(\sqrt{n}(\hat a_n-a), \ \sqrt{n}(\hat b_n-b)\right)^T \rightsquigarrow Z, \ Z\sim\mathcal{N}_2(0,\Sigma), \label{chisq_01}
\end{equation}
\Bin where $\sigma_1^2=\Sigma_{1,1}$, $\sigma_2^2=\Sigma_{2,2}$ and $\sigma_{12}=\Sigma_{1,2}$. From usual properties of Gaussian vectors, we have that, whenever
$det(\Sigma)=\sigma_1^2\sigma_2^2-\sigma_{12}^2\neq 0$,
$$
Z^T \Sigma^{-1} Z \sim \chi_{2}^{2}.
$$
\Bin (See for example \cite{ips-mfpt-ang}, Proposition 12, page 150). By the continuous mapping theorem (see for example \cite{ips-wcia-ang}, Proposition 03, page 34 ), we will have, as $n\rightarrow +\infty$,
$$
Q_n=\left(\sqrt{n}(\hat a_n-a), \ \sqrt{n}(\hat b_n-b)\right) \Sigma^{-1} \left(\sqrt{n}(\hat a_n-a), \ \sqrt{n}(\hat b_n-b)\right)^T \rightsquigarrow \chi_{2}^{2},
$$
\Bin which, as $n\rightarrow +\infty$, leads to
\begin{equation}
Q_n=\frac{n}{\sigma_1^2\sigma_2^2-\sigma_{12}^2} \left[\sigma_2^2(\hat a_n-a)^2+\sigma_1^2(\hat b_n-b)^2 - 2\sigma_{12} (\hat a_n-a)(\hat b_n-b)\right] \rightsquigarrow \chi_{2}^{2}. \label{chisq_02}
\end{equation}
\Bin So, below, for each treated case, we will state two results according to \eqref{chisq_01} and \eqref{chisq_02}.
\section{Asymptotics related to moments estimators} \label{sec2}
\Ni In that section, we are going to treat the following probability laws:
$$
(1) X \sim \gamma(a,b), \ \ (2) X \sim \beta (a,b) , \ \ (3) X \sim \mathcal{U} (a,b) \ \ and \ \ (4) X \sim \mathcal{F} (a,b)
$$
\Bin These results are meant to be interesting examples for other cases not handing here. We stress that the techniques in \cite{lofep} will be extensively used in the following.
\subsection{Gamma laws $\gamma(a,b)$ of parameters $a>0$, $b>0$}
\Ni The gamma law $\gamma(a,b)$ has the probability density function \textit{pdf}
$$
f(x)=\frac{b^a}{\Gamma(a)} x^{a-1} \ \exp(-bx) 1_{(x\geq 0)} \ with \ \Gamma(a)=\int_{a}^{+\infty} x^{a-1} \ \exp(-x) \ dx.
$$
\Bin The $k$-th cumulant ($k\geq 1$) is given by
$$
\mathbb{E} X= \frac{a}{b} \ \ \ and \ \ \ \mathbb{E} X^k= \frac{a}{b^k} \prod_{j=1}^{k-1} (a+j) \ for \ k\geq 2,
$$
\Bin and the variance is
$$
\sigma^2=\mathbb{V}ar(X)=\frac{a}{b^2}.
$$
\Bin The moment estimators $\hat a_n$ and $\hat b_n$ are solution of the equations $a/b=\overline{X}_{n}$ and $ab^{-2}=S_n^2$. We get
$$
(\hat a_n, \ \ \hat b_n)=\left(\frac{\overline{X}_{n}^2}{S_n^2}, \ \ \frac{\overline{X}_{n}}{S_n^2}\right).
$$
\Bin Here are the results for the $\gamma$-law of parameters $a>0$ and $b>0$.\\
\begin{theorem}\label{gamma_01} We have
$$
\sqrt{n}(\hat a_n-a, \ \ \hat b_n-b) \rightsquigarrow \mathcal{N}_2(0, \Sigma),
$$
\Bin with
$$
\Sigma_{1,1}=\mathbb{V}ar(H(X)), \ \ \Sigma_{2,2}=\mathbb{V}ar(L(X)), \ \
\Sigma_{1,2}=\mathbb{C}ov(H(X), L(X))
$$
\Bin and
$$
H=\frac{2\mu \left( \sigma^2 +1\right) }{\sigma ^{4}}h_{1}+\frac{\mu ^{2}}{\sigma ^{4}}h_{2},
$$
$$
L=\left( \frac{\sigma ^{2}+2\mu }{\sigma ^{4}}\right) h_{1}-\frac{\mu }{\sigma ^{4}}h_{2}.
$$
\Bin We also have
\begin{equation}
Q_n=\frac{n}{det(\Sigma)} \left [\Sigma_{2,2}(\hat a_n-a)^2+\Sigma_{1,1}(\hat b_n-b)^2 - 2\Sigma_{1,2} (\hat a_n-a)(\hat b_n-b) \right] \rightsquigarrow \chi_{2}^{2}. \label{chisq_02gamma}
\end{equation}
\end{theorem}
\subsection{ Beta law $\beta(a,b)$ of parameters $a>0$, $b>0$}
\Ni The Beta law has the following probability distribution function
\begin{equation*}
f\left(x\right) =\frac{x^{a-1}\left(1-x\right) ^{b-1}}{B\left( a,b\right) } ,x> 0.
\end{equation*}
\Bin Where
\begin{equation*}
B\left(a,b\right) =\frac{\Gamma \left( a\right) \Gamma \left( b\right) }{\Gamma \left( a+b\right) }.
\end{equation*}
\Bin The expectation is given by
\begin{equation*}
\mathbb{E}\left( X\right) =\frac{a}{a+b}
\end{equation*}
\Bin and the second moment order cumulant is given by
\begin{equation*}
\mathbb{E}\left(X^2 \right) =\frac{a(a+1)}{\left( a+b\right)\left(a+b+1\right) }.
\end{equation*}
\Bin The moment estimators $\hat a_n$ and $\hat b_n$ are solutions of the equations $a/(a+b)=\overline{X}_{n}$ and $a(a+1)/(a+b)(a+b+1)=\overline{{X}_{n}^2}$. We get
$$
(\hat a_n, \ \ \hat b_n)=\left(\frac{\overline{X}_{n}\left(\overline{X}_{n}-\overline{{X}_{n}^2}\right)}{\overline{{X}_{n}^2}-\overline{X}_{n}^2 }, \ \ \frac{\left(1-\overline{X}_{n}\right)\left(\overline{X}_{n}-\overline{{X}_{n}^2}\right)}{\overline{{X}_{n}^2}-\overline{X}_{n}^2}\right)
$$
\Bin Here are the results for the $\beta$-law of parameters $a>0$ and $b>0$.\\
\begin{theorem}\label{asympt_beta_01} We have
$$
\sqrt{n}(\hat a_n-a, \ \ \hat b_n-b) \rightsquigarrow \mathcal{N}_2(0, \Sigma),
$$
\Bin with
$$
\Sigma_{1,1}=\mathbb{V}ar(H(X)), \ \ \Sigma_{2,2}=\mathbb{V}ar(L(X)), \ \
\Sigma_{1,2}=\mathbb{C}ov(H(X), L(X)),
$$
$$
H=\frac{\sigma^{2}\left( 2\mu -m_{2}\right) +2\mu^2 \left( \mu-m_{2}\right) }{\sigma ^{4}}h_{1}-\frac{\sigma^{2}\mu + \mu \left( \mu -m_{2}\right)}{\sigma^{4}}h_{2}
$$
\Bin and
$$
L=\frac{\sigma^{2}\left( m_{2} - 2\mu+ 1 \right)+ 2\mu(1-\mu)(\mu -m_{2} )}{\sigma^{4}}h_{1}+\frac{\left( \mu - 1 \right) \left( \sigma^{2}+ \mu -m_{2}\right)}{\sigma^{4}}h_{2}.
$$
\Bin We also have the following asymptotic $\chi^2$ result
\begin{equation}
Q_n=\frac{n}{det(\Sigma)} \left[\Sigma_{2,2}(\hat a_n-a)^2+\Sigma_{1,1}(\hat b_n-b)^2 - 2\Sigma_{1,2} (\hat a_n-a)(\hat b_n-b)\right] \rightsquigarrow \chi_{2}^{2}. \label{chisq_02gamma}
\end{equation}
\end{theorem}
\subsection{ The Uniform law of parameters $\mathcal{U}(a,b)$ , $a>0$ and $b>a$}
\Ni The probability distribution function of the uniform law is given by
\begin{equation*}
f\left(x\right) =\frac{1}{b-a}, x\in \left[a,b\right].
\end{equation*}
\Bin Its expectation is
\begin{equation*}
\mathbb{E}\left(X\right) =\frac{a+b}{2}.
\end{equation*}
\Bin The variance is
\begin{equation*}
\mathbb{V}ar\left(X\right) =\frac{\left(b-a\right) ^{2}}{12}.
\end{equation*}
\Bin The moment estimators are the solutions of the equations
$$
(\hat a_n, \ \ \hat b_n)=\left(\overline{X}_{n}-\lambda \left( S_{n}^{2}\right) ^{1/2}, \ \ \overline{X}_{n}+\lambda \left(S_{n}^{2}\right) ^{1/2}\right)
$$
\Bin where $\lambda=12^{1/2}/2$.
\Bin Here are the results for the Uniform-law.\\
\begin{theorem}\label{asympt_Uniform_01} We have
$$
\sqrt{n}(\hat a_n-a, \ \ \hat b_n-b) \rightsquigarrow \mathcal{N}_2(0, \Sigma),
$$
\Bin with
$$
\Sigma_{1,1}=\mathbb{V}ar(H(X)), \ \ \Sigma_{2,2}=\mathbb{V}ar(L(X)), \ \
\Sigma_{1,2}=\mathbb{C}ov(H(X), L(X)),
$$
\Bin where
$$
H=\left( 1+\frac{\lambda\mu }{\sigma }\right) h_{1}-\frac{\lambda }{2\sigma }h_{2}
$$
\Bin and
$$
L=\left( 1-\frac{\lambda\mu }{\sigma }\right) h_{1}+\frac{\lambda }{2\sigma }h_{2},
$$
\Bin where $\lambda=12^{1/2}/2$.\\
\Bin We also have
\begin{equation}
Q_n=\frac{n}{det(\Sigma)} \left[\Sigma_{2,2}(\hat a_n-a)^2+\Sigma_{1,1}(\hat b_n-b)^2 - 2\Sigma_{1,2} (\hat a_n-a)(\hat b_n-b) \right] \rightsquigarrow \chi_{2}^{2}. \label{chisq_02gamma}
\end{equation}
\end{theorem}
\subsection{ Fisher law $\mathcal{F}(a,b)$ of parameters $a>0$ and $b>0$}
\Ni For a Fisher law with $a$ and $b$ degrees of freedom, the parameters are supposed to be integers. But in the general case, the probability density function has the same form and is associated to the quotient of two independent random variables $Z_1/Z_2$, where $Z_1 \sim \gamma(a,1/2)/a$ and $Z_2 \sim \gamma(b,1/2)/b$, where $a$ and $b$ are positive. The \textit{pdf} is expressed as follows:
\begin{equation*}
f\left(x\right) =\frac{a^{1/2}b^{1/2}\Gamma \left(\frac{a+b}{2}\right)
x^{a/2-1}}{\Gamma \left(a/2\right) \Gamma \left(b/2\right) \left(b+ax\right) ^{\left(a+b\right) /2}},x>0.
\end{equation*}
\Bin The expectation is given by
\begin{equation*}
\mathbb{E}\left(X\right) =\frac{b}{b-2}=\overline{X_{n}}.
\end{equation*}
\Bin The variance is given by
\begin{equation*}
\mathbb{V}ar\left(X\right) =\frac{2b^{2}\left(a+b-2\right) }{a\left(b-2\right) ^{2}\left(b-4\right) }=S_{n}^{2}, \ b>4.
\end{equation*}
\Bin The moment estimators are solutions of the equations
\begin{equation*}
\left( \hat{a}_{n},\ \ \hat b_{n}\right) =\left(\frac{2\overline{X}_{n}^{2}}{S^2(2-\overline{X}_{n})-\overline{X}_{n}^{2}(\overline{X}_{n}-1)}, \ \ \frac{2\overline{X}_{n}}{\overline{X}_{n}-1}\right).
\end{equation*}
\Bin Here are the results for the Fisher-law.\\
\begin{theorem}\label{asympt_Fisher_01} We have
$$
\sqrt{n}(\hat a_n-a, \ \ \hat b_n-b) \rightsquigarrow \mathcal{N}_2(0, \Sigma),
$$
\Bin with
$$
\Sigma_{1,1}=\mathbb{V}ar(H(X)), \ \ \Sigma_{2,2}=\mathbb{V}ar(L(X)), \ \
\Sigma_{1,2}=\mathbb{C}ov(H(X), L(X)),
$$
\Bin where
$$
H=\frac{2\mu(2-\mu)}{\beta}h_{1}-\frac{2\mu^{2}(2-\mu)}{\beta^{2}}h_{2}
$$
\Bin with
$$
\beta=\sigma^{2}+2\mu(2-\mu)-\mu(3\mu-2)
$$
\Bin and where
$$
L=-\frac{2}{(\mu -1)^2}h_{1}.
$$
\Bin We also have
\begin{equation}
Q_n=\frac{n}{det(\Sigma)} \left[ \Sigma_{2,2}(\hat a_n-a)^2+\Sigma_{1,1}(\hat b_n-b)^2 - 2\Sigma_{1,2} (\hat a_n-a)(\hat b_n-b)\right]\rightsquigarrow \chi_{2}^{2}. \label{chisq_02gamma}
\end{equation}
\end{theorem}
\section{Simulations} \label{sec3}
\Bin We are going to describe our simulation works for one of studied distributions. Next we will explain their outputs and their interpretations. Finally, we will display results for all cases. Important scripts will be posted in the appendix \pageref{script_01}.\\
\subsection{Simulation works}. In all cases, we estimate two parameters. In the case of the $\gamma(a,b)$ law, the moment estimators are denoted by \textit{achap} and and \textit{bchap}. We will have three parts.\\
\Ni \textbf{A- Computing the exact moments and other coefficients of the estimators}.\\
\Ni (a) Before proceeding to the Monte-Carlo method, we have to computed the function $H$ and $L$, demoted as \textit{bigH} and \textit{bigL}.\\
\Ni (b) We proceed to numerical methods for computing $\mathbb{E}H(X)$, $sigmaHexaC=\mathbb{V}ar(E)H(X)$, $\mathbb{E}L(X)$,
$sigmaLexaC=\mathbb{V}ar(E)L(X)$ and the exact co-variance $SigmaHLexa=\mathbb{C}ov(H(X),L(X)$, where $X$ stands for random variable with the studied law (here a $\gamma(a,b)$ law). The trapezoidal method algorithm is used for all integral computations here.\\
\Ni In page \pageref{script_01}, the related script is given under the title \textbf{A1 - Computing exact coefficients} . \Bin Table \label{tab1} gives exact values of the variances for different pairs $(a,b)$.\\
\begin{table}
\centering
\label{tab1}
\caption{Over/under estimation of variances and covariances of estimators with respect to the true values for $(a,b)=(2,3))$}
\begin{tabular}{l|lll}
\hline \hline
(a,b) & (2,3) & (3,10)& (10,3)\\
\hline \hline
\textit{sigmaHexaC} & $7.78$ & $107.08$ & $63.08$ \\
\textit{sigmaLexaC} & $7.37$ & $76.58$ & $16.99$ \\
\textit{sigmaHLexa} & $40.3$ & $7.985.01$ & $1006.95$ \\
\textit{Correlation} & $69.76\%$ & $98.11\%$ &$93.95\%$\\
\hline \hline
\end{tabular}
\end{table}
\Ni \textbf{B- Monte-Carlo estimation}.\\
\Ni (a) Fix a sample size $n\geq 2$. Fix values to $a$ and $b$.\\
\Ni (b) Fix the number of repetitions $B=1000$ (big enough to ensure the stability of outcomes).\\
\Ni (c) At each repetition $j \in \{1,\cdots,B\}$, we generate an sample of $X$ of size $n$. Next
\begin{itemize}
\item[(1)] $DA[j]=\sqrt{n}(achap-a)$
\item[(2)] $DB[j]=\sqrt{n}(bchap-a)$
\item[(3)] $VH[j]=sd(bigH(X))$
\item[(4)] $VL[j]=sd(bigL(X)$
\item[(5)] $VHL[j]=cov(bigH(X),bigL(X)$
\end{itemize}
\Ni In page \pageref{script_02}, the related script is under the title \textbf{A2- Script Monte Carlo works}.\\
\Ni \textbf{C- Computing the empirical moments and other coefficients}.\\
\Ni (a) Now, we have: (1) an estimate of $\Sigma_{1,1}$ by the square of the average of the vector $VH$, (2) an estimate of $\Sigma_{2,2}$ by the square of the average of the vector $VH$ and (3) an estimate of $\Sigma_{1,2}$ by the average of the vector $VHL$. We denote them as \textit{SigmaHEMP}, \textit{SigmaLEMP} and \textit{SigmaHLEMP}.\\
\Ni (b) We also have: (1) an estimate of $\Sigma_{1,1}$ by empirical variance of $DA$, (2) an estimate of $\Sigma_{2,2}$
by empirical variance of $DB$ and (3) an estimate of $\Sigma_{1,2}$ by by empirical covariance between $DA$ and $DB$. We denote them as \textit{sigmaHSAMP}, \textit{sigmaLSAMP} and \textit{sigmaHLSAMP}.\\
\Ni In page \pageref{script_03}, the related script for computing \textit{sigmaHEMP} , \textit{sigmaLEMP}, \textit{sigmaHLEMP}, \textit{sigmaHSAMP}, \textit{sigmaLSAMP} and \textit{sigmaHLSAMP} is given under the title \textbf{A3- Over/under estimations of variances and covariances} from the script \textbf{A2- Script Monte Carlo works} in page \pageref{script_02}.\\
\Ni In Tables \ref{tab2a}, \ref{tab2b} and \ref{tab2c} display the quotients of empirical coefficients over the true coefficients, allowing to over or under-estimation, for three values of pairs $(a,b)$.\\
\begin{table}
\centering
\caption{Over/under estimation of variances and covariances of estimators with respect to the true values for $(a,b)=(2,3))$}
\label{tab2a}
\begin{tabular}{l|llll}
\hline \hline
size & n=50 & n=100 & n=200 & n=1000\\
\hline \hline
\textit{Qsig-1emp} & $98.98\%$ & $98.12\%$ &$98.47\%$ & $98.49\%$ \\
\textit{Qsig-2emp} & $72.00\%$ & $74.59\%$ &$74.85\%$ &$75.56\%$ \\
\textit{Qsig-12emp} & $81\%$ & $81.05\%$ &$82.39\%$ & $81.89\%$ \\
\textit{Qsig-1samp} & $44.03\%$ & $43.34\%$ &$43.57\%$ & $44.93\%$ \\
\textit{Qsig-2samp} & $79.11\%$ & $76.62\%$ &$76.58\%$ &$78.81\%$ \\
\textit{Qsig-12samp} & $46.34\%$ & $44.57\%$ &$45.04\%$ &$48.23\%$ \\
\hline \hline
\end{tabular}
\end{table}
\begin{table}
\centering
\caption{Over/under estimation of variances and covariances of estimators with respect to the true values for $(a,b)=(3,10))$}
\label{tab2b}
\begin{tabular}{l|lllll}
\hline \hline
size & n=50 & n=100 & n=200 & n=1000 \\
\textit{Qsig-1emp} & $96.87\%$ & $97.59\%$ & $98.46\%$ & $98.19\%$ \\
\textit{Qsig-2emp} & $99.44\%$ & $99.95\%$ & $100.49\%$ & $100.32\%$ \\
\textit{Qsig-12emp} & $98.72\%$ & $99.33\%$ & $100.46\%$ & $100.00\%$ \\
\textit{Qsig-1samp} & $4.96\%$ & $4.46\%$ & $4.43\%$ & $4.40\%$ \\
\textit{Qsig-2samp} & $25.03\%$ & $22.34\%$ & $21.98\%$ & $22.12\%$ \\
\textit{Qsig-12samp} & $1.199\%$ & $0.96\%$ & $0.93\%$ & $0.94\%$ \\
\hline \hline
\end{tabular}
\end{table}
\begin{table}
\centering
\caption{Over/under estimation of variances and covariances of estimators with respect to the true values for $(a,b)=(10,3))$}
\label{tab2c}
\begin{tabular}{l|lllll}
\hline \hline
size & n=50 & n=100 & n=200 &n=1000\\
\textit{Qsig-1emp} & $91.58\%$ & $92.92\%$ & $94.29\%$ & $94.25\%$ \\
\textit{Qsig-2emp} & $90.25\%$ & $91.77\%$ & $93.30\%$ & $94.29\%$ \\
\textit{Qsig-12emp} & $85.72\%$ & $86.95\%$ & $88.93\%$ & $88.14\%$ \\
\textit{Qsig-1samp} & $25.09\%$ & $23.26\%$ & $23.79\%$ & $23.54\%$ \\
\textit{Qsig-2samp} & $30.30\%$ & $28.15\%$ & $28.19\%$ & $28.37\%$ \\
\textit{Qsig-12samp} & $7.43\%$ & $6.404\%$ & $6.56\%$ & $6.54\%$ \\
\hline \hline
\end{tabular}
\end{table}
\Ni \textbf{D- Statistical tests for Computing the empirical moments and other coefficients}.\\
\Ni (1) Performance of the point estimation. From the script \textbf{A2- Script Monte Carlo works} in page \pageref{script_02}, we can compute the mean error (ME), the mean absolute error (MAE) and the square-root of the mean square error (MSE)of the point estimations on $a$ and $b$ the R codes \textit{mean(DACHAP-a)},
\textit{mean(DBCHAP-b)}, \textit{mean(abs(DACHAP-a))}. \textit{mean(abs(DBCHAP-b))}, \textit{sd(DACHAP-a)} and \textit{sd(DBCHAP-b)}. We report their values in Table \ref{tabpe}
\begin{table}
\centering
\begin{tabular}{cccccccc}
Error type & n=25 & n=50 & n=75 & n=100 & n=200 & n=300 & n=1000\\
\hline \hline
ME (A) & 1.09 & 0.53& 0.33 &0.13 & 0.13 & 0.08 &0.02\\
MAE (A) & 2.88 & 1.99 & 1.54 & 0.92 & 0.92 & 0.76 &0.411\\
$\sqrt{MSE}$ (A) & 3.88 &2.49 & 1.99 & 1.163 & 1.17 & 0.95 &0.5\\
\hline \hline
ME (B) & 0.42 & 0.21 & 0.12 & 0.05 & 0.048 & 0.029 &0.009\\
MAE (B) & 1.07 & 0.72 & 0.57 & 0.34 & 0.034 & 0.28 &0.15\\
$\sqrt{MSE}$ (B)&1.44 & 0.92&0.73 & 0.43 & 0.43 & 0.35 &0.19\\
\hline \hline
\end{tabular}
\caption{Evolution of the errors on estimation of $a=10$ and $b=3$ in the sample sizes}
\label{tabpe}
\end{table}
\Ni (2) Statistical tests on $a$. We have three tools
\begin{eqnarray*}
&&DA_1[j]=\sqrt{\frac{n}{SigmaHexaC}} (achap-a)\approx \mathcal{N}(0,1),\\
&&DA_2[j]=\sqrt{\frac{n}{SigmaHEMP}} (achap-a) \approx \mathcal{N}(0,1),\\
&&DA_3[j]=\sqrt{\frac{n}{SigmaHSAMP}} (achap-a) \approx \mathcal{N}(0,1).
\end{eqnarray*}
\Ni (3) Statistical tests on $b$. We have three tools
\begin{eqnarray*}
&&DB_1[j]=\sqrt{\frac{n}{SigmaLexaC}} (bchap-b) \approx \mathcal{N}(0,1),\\
&&DB_2[j]=\sqrt{\frac{n}{SigmaLEMP}} (bchap-b)\approx \mathcal{N}(0,1),\\
&&DB_3[j]=\sqrt{\frac{n}{SigmaLSAMP}} (bchap-b) \approx \mathcal{N}(0,1).
\end{eqnarray*}
\Ni In page \pageref{script_04}, the scripts \textbf{A4- Computations of the p-values}, for each parameter $a$ and $b$, we compute the empirical p-values for each sequence, as the frequency of element of the sequence exceeding $1.96$. The test is satisfactory if that $p$ value is less of around $5\%$. The different p-values for $a=2$ and $b=3$ are given for different values of $n$ in Table \ref{tab3c}.
\begin{table}
\centering
\caption{Empirical p-values for statistical test of $a$ and $b$ ($a=10$ and $b=3$)}
\label{tab3c}
\begin{tabular}{l|lllll}
\hline \hline
cases & n=50 & n=100 & n=150 &n=200 &n=1000\\
Exact & $0\%$ & $0\%$ & $0\%$ & $0\%$ & $0\%$ \\
Empirical & $0\%$ & $0\%$ & $0\%$ & $0\%$ & $0\%$ \\
Sample & $5.4\%$ & $5.3\%$ & $5.18\%$ & $5.09\%$ & $5\%$ \\
Exact & $0.01\%$ & $0\%$ & $0\%$ & $0\%$ & $0\%$ \\
Empirical & $0.02\%$ & $0\%$ & $0\%$ & $0\%$ & $0\%$ \\
Sample & $5.43\%$ & $5.49\%$ & $5.12\%$ & $5.28\%$ & $5.28\%$ \\
\hline \hline
\end{tabular}
\end{table}
\Ni To test the quality of the normal approximations, we display the QQ-plots and the Parzen estimators graphs for each parameter in Fig
\ref{fig3a50} (QQ-plots and Parzen estimators related to the parameter $a$ for n=50, according to the type of estimation of the coefficients),
in Fig \ref{fig3b50} (QQ-plots and Parzen estimators related to the parameter $b$ for n=50, according to the type of estimation of the coefficients),in Fig \ref{fig3a300} (QQ-plots and Parzen estimators related to the parameter $a$ for n=300, according to the type of estimation of the coefficients) and in Fig \ref{fig3b300} (QQ-plots and Parzen estimators related to the parameter $b$ for n=300, according to the type of estimation of the coefficients) in Appendix C
(Page \pageref{appendixC}).\\
\Bin \textbf{(E) Omnibus test}. We mean by omnibus test that the combine both test into a chi-square test as in Part (b) of each of Theorems \ref{asympt_Fisher_01}, \ref{asympt_Uniform_01}, \ref{asympt_beta_01} \label{gamma_01}. Depending on the use of exact values, empirical values or sample values of the variance of co-variances, we have
three statistics that can be used each for the chi-square test:
\begin{eqnarray*}
\text{QExa}[j]&=&\frac{n}{detHL} \biggr(sigmaLExaC(achap[j]-a)^2\\
&+&sigmaHexaC(bchap[j]-b)^2\\
&-& 2\times sigmaHLexa(achap[j]-a)\times (bchap[j]-b)\biggr) \sim \chi_1^2\\
\text{QEMPDB}[j]&=&\frac{n}{detHLEMP} \biggr((sigmaLEMP^2)(achap[j]-a)^2\\
&+&(sigmaHEMP^2)(bchap[j]-b)^2\\
&-& 2\times sigmaHLEMP(achap[j]-a)\times (bchap[j]-b)\biggr) \sim \chi_1^2\\
\text{QEMPDB}[j]&=&\frac{n}{detHLSAMP} \biggr((sigmaLSAMP^2)(achap[j]-a)^2\\
&+&(sigmaHSAMP^2)(bchap[j]-b)^2\\
&-&2 \times sigmaHLSAMP(achap[j]-a)\times (bchap[j]-b)\biggr) \sim \chi_1^2
\end{eqnarray*}
\Bin Table \ref{tab4} provides the p-value related to the omnibus test for different sizes according to the estimations of the coefficients used. The related test is given in the script under the title \textit{A5- p-values for the omnibus statistical test} in page \pageref{script_05}.
\begin{table}
\centering
\caption{QNE, QNEMP and QNSAMP according to the size $n$ for $a=10$ and $b=3$}
\label{tab4}
\begin{tabular}{l|llllll}
\hline \hline
cases & n=50 & n=100 &n=200 &n=300 &n=1000\\
QNE & $1.2\%$ & $0.1\%$ & $0.02\%$ & $0.001\%$ & $0.01\%$ \\
QNEMP & $0\%$ & $0\%$ & $0\%$ & $0\%$ & $0.8\%$ \\
QNSAMP & $0.6\%$ & $0.1\%$ & $0.3\%$ & $0.01\%$ & $0.2\%$ \\
\hline \hline
\end{tabular}
\end{table}
\Bin \textbf{(D) - Conclusions of recommendations from simulations}. The simulation studies show that the omnibus statistical test is very good even for sizes as small as $n=50$ for all estimations of the coefficients in the test statistics. When we do Gaussian separate tests for $a$ and $b$, the outcomes are remarkable in the use the variance and covariance of $\sqrt{n}(\hat a_n-a)$ and $\sqrt{n}(\hat b_n-b)$. This is observable in the QQ-plots, the Parzen graphs and in the p-values of the tests. The separate tests seem to recommend the tests when $n$ is bigger than $100$. But, definitively, the omnibus works fine for small sizes as $n=11$ with p-values $5.6\%$, $0\%$, $1.7\%$.\\
\Ni We strongly suggest to no use the tests with empirical estimations of the variance and covariance which lead to severe under or over estimation.
\section{Proofs of Theorems} \label{sec4}
\Ni Here, we provide the computations for each treated probability law.
\subsection{ Gamma Law $ \gamma(a,b) $ of parameters $a>0$, $b>0$ }
\Ni We have
\begin{equation}\label{gamma_01}
\overline{X}_{n} =\mu +n^{-1/2}G_{n}\left( h_{1}\right)
\end{equation}
\Bin and
\begin{eqnarray}\label{gamma_02}
S_{n}^{2} &=&\frac{1}{n-1}\left[\sum_{j=1}^{n}X_{j}^{2}-n\overline{X}_n^{2}\right] \\
&=&\frac{n}{n-1}\left[ \frac{1}{n}\sum_{j=1}^{n}X_{j}^{2}-\overline{X}_{n}^{2}\right]\\
&=& \frac{n}{n-1}\left[\overline{X_{n}^{2}} - \overline{X}_{n}^{2}\right].
\end{eqnarray}
\Bin By the delta method,
\begin{equation}\label{gamma_03}
\overline{X}_n^{2} =\mu ^{2}+n^{-1/2}G_{n}\left( 2\mu h_{1}\right) +O_{p}\left(n^{-1/2}\right)
\end{equation}
\Bin and
\begin{equation}\label{gamma_04}
\overline{X_{n}^{2}}=\frac{1}{n}\sum_{j=1}^{n}X_{j}^{2} =m_{2}+n^{-1/2}G_{n}\left(h_{2}\right).
\end{equation}
\Bin We know that
\begin{equation*}
\frac{n}{n-1}=\left( \frac{n-1}{n}\right) ^{-1}=\left( 1-1/n\right)^{-1}=1+O_{p}\left( 1\right).
\end{equation*}
\Bin Hence
\begin{equation}\label{gamma_05}
S_{n}^{2}=\sigma ^{2}+n^{-1/2}G_{n}\left( h_{2}-2\mu h_{1}\right)+O_{p}\left( n^{-1/2}\right).
\end{equation}
\Bin Let us handle $\hat{b}_{n}$. We have
\begin{equation*}
\hat b_{n}=\frac{\overline{X_{n}}}{S_{n}^{2}}.
\end{equation*}
\Bin By equations \eqref{gamma_01}, \eqref{gamma_05} and by lemma $11$ in \cite{lofep}, we have
\begin{eqnarray*}
\hat{b_{n}} &=&\frac{\mu }{\sigma ^{2}}+n^{-1/2}G_{n}\left( \frac{h_{1}}{\sigma
^{2}}-\frac{\mu }{\sigma ^{4}}\left( h_{2}-2\mu h_{1}\right) \right)+O_{p}\left( n^{-1/2}\right) \\
&=&\frac{\mu }{\sigma ^{2}}+n^{-1/2}G_{n}\left( L\right),
\end{eqnarray*}
\Bin where
\begin{equation*}
L=\frac{h_{1}}{\sigma ^{2}}-\frac{\mu }{\sigma ^{4}}\left( h_{2}-2\mu h_{1}\right) =\left( \frac{\sigma ^{2}+2\mu h_{1}}{\sigma ^{4}}\right) h_{1}-\frac{\mu }{\sigma ^{4}}h_{2}
\end{equation*}
\Bin Let us treat $\hat a_{n}$. By combining equations $\eqref{gamma_03}$, $\eqref{gamma_05}$ and lemma $11$ in \cite{lofep} , we get
\begin{equation*}
\hat a_{n}=\frac{\mu }{\sigma ^{2}}+n^{-1/2}G_{n}\left(H\right),
\end{equation*}
\Bin where
\begin{eqnarray*}
H &=&\frac{2\mu }{\sigma ^{2}}h_{1}-\frac{\mu ^{2}}{\sigma ^{4}}\left(h_{2}-2\mu h_{1}\right) \\
&=&\frac{2\mu \sigma ^{2}+2\mu }{\sigma ^{4}}h_{1}-\frac{\mu ^{2}}{\sigma
^{4}}h_{2} \\
&=&\frac{2\mu \left( \sigma^2 +1\right) }{\sigma ^{4}}h_{1}-\frac{\mu ^{2}}{\sigma ^{4}}h_{2}.
\end{eqnarray*}
\subsection{Beta Law $ \beta (a,b) $ of parameters $a>0$, $b>0 $}
\Ni The moment estimators $\hat a_n$ and $\hat b_n$ are solutions of the equations
\begin{equation*}
\hat{a}_{n}=\frac{\overline{X}_{n}\left( \overline{X}_{n}-\overline{X_{n}^{2}}\right)}{\overline{X_{n}}^{2}-\overline{X_{n}^{2}}}
\end{equation*}
\Bin and
\begin{equation*}
\hat b_{n}=\frac{\left( 1-\overline{X}_{n}\right) \left( \overline{X}_{n}-\overline{X_{n}^{2}}\right) }{\overline{X_{n}}^{2}-\overline{X_{n}^{2}}}.
\end{equation*}
\Bin We have
\begin{equation} \label{beta_01}
\overline{X}_{n} =\mu +n^{-1/2}G_{n}\left( h_{1}\right)
\end{equation}
\Bin and
\begin{equation} \label{beta_02}
\overline{X_{n}^{2}} =m_{2}+n^{-1/2}G_{n}\left( h_{2}\right).
\end{equation}
\Bin By the delta method, we have
\begin{equation} \label{beta_03}
\overline{X_{n}}^2 =\mu ^{2}+n^{-1/2}G_{n}\left( 2\mu h_{1}\right)
+O_{p}\left(n^{-1/2}\right)
\end{equation}
\Bin and
\begin{eqnarray} \label{beta_04}
1-\overline{X}_{n}&=&1-\mu -n^{-1/2}G_{n}\left( h_{1}\right) \notag \\
&=&1-\mu +n^{-1/2}G_{n}\left( -h_{1}\right).
\end{eqnarray}
\Bin By combining equations \eqref{beta_01} and \eqref{beta_02}, we have
\begin{equation*}
A_{n}=\overline{X}_{n}-\overline{X_{n}^{2}}=\mu -m_{2}+n^{-1/2}G_{n}\left(h_{1}-h_{2}\right).
\end{equation*}
\Bin By combining equations \eqref{beta_02} and \eqref{beta_03}, we have
\begin{eqnarray*}
B_{n} &=&\overline{X_{n}^2}-\overline{X_{n}}^2=m_{2}-\mu
^{2}+n^{-1/2}G_{n}\left(h_{2}\right) -n^{1/2}G_{n}\left( 2\mu h_{1}\right) \\
&=&m_{2}-\mu ^{2}+n^{-1/2}G_{n}\left( h_{2}-2\mu h_{1}\right)
+O_{p}\left(n^{-1/2}\right).
\end{eqnarray*}
\Bin Hence
\begin{equation*}
\hat a_{n}=\frac{C_{n}}{B_{n}}
\end{equation*}
\Bin with
\begin{eqnarray*}
C_{n} &=&\overline{X}_{n}\times A_{n} \\
&=&\mu \left( \mu -m_{2}\right) +n^{-1/2}G_{n}\left( \left( \mu h_{1}-\mu h_{2}\right) +\left( \mu -m_{2}\right) h_{1}\right) +O_{p}\left(
n^{-1/2}\right).
\end{eqnarray*}
\Bin Then
\begin{equation*}
C_{n}=\mu \left( \mu -m_{2}\right) +n^{-1/2}G_{n}\left(H_{2}\right)+O_{p}\left( n^{-1/2}\right),
\end{equation*}
\Bin where
\begin{eqnarray*}
H_{2} &=&\left( \mu h_{1}-\mu h_{2}\right) +\left( \mu -m_{2}\right) h_{1} \\
&=&\left( 2\mu -m_{2}\right) h_{1}-\mu h_{2}.
\end{eqnarray*}
\Bin Hence
\begin{equation*}
\hat a_{n}=\frac{\mu \left( \mu -m_{2}\right) +n^{-1/2}G_{n}\left(H_{2}\right)+O_{p}\left( n^{-1/2}\right) }{m_{2}-\mu
^{2}+n^{-1/2}G_{n}\left( h_{2}-2\mu h_{1}\right) +O_{p}\left(n^{-1/2}\right) }.
\end{equation*}
\Bin By lemma $11$ in \cite{lofep}, we get
\begin{equation*}
\hat a_{n}=\frac{\mu \left( \mu -m_{2}\right) }{m_{2}-\mu ^{2}}+n^{-1/2}G_{n}\left( H\right) +O_{p}\left( n^{-1/2}\right),
\end{equation*}
\Bin Where
$$
H=\frac{\sigma^{2}\left( 2\mu -m_{2}\right) +2\mu^2 \left( \mu-m_{2}\right) }{\sigma ^{4}}h_{1}-\frac{\sigma^{2}\mu + \mu \left( \mu -m_{2}\right)}{\sigma^{4}}h_{2}.
$$
\Bin Let us handle now $\hat b_{n}$ .\newline
\Ni Remind that
\begin{equation*}
\hat{b}_{n}=\frac{\left( 1-\overline{X}_{n}\right) \left( \overline{X}_{n}-X_{n}^{2}\right) }{\overline{X_{n}^{2}}-\overline{X}_{n}^{2}}=\frac{B_{1}\left(
n\right) }{B_{2}\left( n\right) }.
\end{equation*}
\Bin Thanks to the previous calculus, we have
\begin{equation*}
\hat{b}_{n}=\frac{( 1-\mu )(\mu - m_{2})}{\sigma^{2}} +n^{-1/2}G_{n}\left( L\right) +O_{p}\left(n^{-1/2}\right),
\end{equation*}
\Bin where
$$
L=\frac{\sigma^{2}\left( m_{2} - 2\mu+ 1 \right)+ 2\mu(1-\mu)(\mu -m_{2} )}{\sigma^{4}}h_{1}+\frac{\left( \mu - 1 \right) \left( \sigma^{2}+ \mu -m_{2}-\right)}{\sigma^{4}}h_{2}.
$$
\subsection{Uniform Law $\mathcal{U}(a,b)$, of parameters $a>0$ and $b>a$}
\Ni We have
\begin{equation} \label{beta_01}
\overline{X}_{n}=\mu +n^{-1/2}G_{n}\left(h_{1}\right)
\end{equation}
\Bin and
\begin{equation} \label{beta_02}
\overline{X_{n}^{2}} =m_{2}+n^{-1/2}G_{n}\left( h_{2}\right).
\end{equation}
\Bin By combining equations \eqref{beta_01} and \eqref{beta_02}, we have
\begin{equation*}
S_{n}^{2}=\sigma ^{2}+n^{-1/2}G_{n}\left( h_{2}-2\mu h_{1}\right).
\end{equation*}
\Bin By the delta method, we have
\begin{equation*}
\left( S_{n}^{2}\right) ^{1/2}=\sigma +n^{-1/2}G_{n}\left( \frac{1}{2\sigma}\left( h_{2}-2\mu h_{1}\right) \right) +O_{p}\left(n^{-1/2}\right).
\end{equation*}
\Bin So
\begin{eqnarray*}
a_{n} &=&\mu -\lambda \sigma +n^{-1/2}G_{n}\left( \left( h_{1}-\frac{\lambda}{2\sigma}\left( h_{2}-2\mu h_{1}\right) \right) \right) +O_{p}\left(n^{-1/2}\right) \\
&=&\mu -\lambda \sigma +n^{-1/2}G_{n}\left( L\right) +O_{p}\left(n^{-1/2}\right),
\end{eqnarray*}
\Bin with
\begin{equation*}
L=\left( 1+\frac{\lambda\mu }{\sigma}\right) h_{1}-\frac{\lambda }{2\sigma}h_{2}.
\end{equation*}
\Bin By using the same technique, we have
\begin{equation*}
\hat b_{n}=\mu -\lambda \sigma+n^{-1/2}G_{n}\left( H\right) +O_{p}\left(n^{-1/2}\right),
\end{equation*}
\Bin where
\begin{equation*}
H=\left( 1-\frac{\lambda\mu }{\sigma}\right) h_{1}+\frac{\lambda }{2\sigma}h_{2}.
\end{equation*}
\subsection{Fisher Law $\mathcal{F}(a,b)$ of parameters $a$ and $b$}
\Bin The moment estimators are defined below. The first moment estimator is
\begin{eqnarray}\label{fisher_01}
\hat{b}&=&\frac{2\overline{X}_{n}}{\overline{X}_{n}-1}\\
\notag &=&\frac{2\mu+n^{-1/2}\mathbb{G}_{n}(2h_{1})}{\mu-1+n^{-1/2}\mathbb{G}_{n}(-h_{1})}\\
\notag &=&\frac{2\mu}{\mu-1}+n^{-1/2}\mathbb{G}_{n}(L)+o_{\mathbb{P}}(n^{-1/2}),
\end{eqnarray}
\Bin where
\begin{eqnarray*}
L&=&\frac{2h_{1}}{\mu -1}-\frac{2\mu}{(\mu-1)^2}h_{1}\\
&=&-\frac{2}{(\mu -1)^2}h_{1}.
\end{eqnarray*}
\Bin The second estimator is given by
$$
a=\frac{2b^{2}(b-2)}{S^2(b-2)^2(b-4)-2b^4}.
$$
\Bin By equation \eqref{fisher_01}, we know that
$$
b=\frac{2\overline{X}_{n}}{\overline{X}_{n}-1}.
$$
\Bin So we have,
\begin{eqnarray*}
b-2&=&\frac{2\overline{X}_{n}}{\overline{X}_{n}-1}-2=\frac{2}{\overline{X}_{n}-1},\\
b-4&=&\frac{2\overline{X}_{n}}{\overline{X}_{n}-1}-4=\frac{-2\overline{X}_{n}+4}{\overline{X}_{n}-1}.
\end{eqnarray*}
\Bin Hence
\begin{eqnarray*}
\hat{a}&=&\frac{16\overline{X}_{n}^{2}}{8S^2(2-\overline{X}_{n})-8\overline{X}_{n}^{2}(\overline{X}_{n}-1)}\\
&=&\frac{2\overline{X}_{n}^{2}}{S^2(2-\overline{X}_{n})-\overline{X}_{n}^{2}(\overline{X}_{n}-1)}.
\end{eqnarray*}
\Bin From \eqref{fisher_01}, we have
\begin{equation}\label{fisher_02}
S_{n}^{2}=\sigma^{2}+n^{-1/2}\mathbb{G}_{n}(h_{2}-2\mu h_{1})+o_{\mathbb{P}(n^{-1/2})}.
\end{equation}
\Bin Further, we have
\begin{equation}\label{fisher_03}
2-\overline{X}_{n}=2-\mu +n^{-1/2}\mathbb{G}_{n}(-h_{1})
\end{equation}
\begin{equation}\label{fisher_04}
S_{n}^{2}(2-\overline{X}_{n})=\sigma^{2}(2-\mu)+n^{-1/2}\mathbb{G}_{n}(L_{1})+o_{\mathbb{P}(n^{-1/2})},
\end{equation}
\Bin where
\begin{eqnarray*}
L_{1}&=&-\sigma^{2}h_{1}+(2-\mu)(h_{2}-2\mu h_{1})\\
&=&(\sigma^{2}+2\mu(2-\mu))h_{1}+(2-\mu)h_{2}
\end{eqnarray*}
\Bin and
$$
\overline{X}_{n}^{2}=\mu^{2}+n^{-1/2}\mathbb{G}_{n}(2\mu h_{1})+o_{\mathbb{P}(n^{-1/2})}.
$$
\Bin We also have
\begin{equation}\label{fisher_05}
\overline{X}_{n}-1=\mu -1 +\mathbb{G}_{n}(h_{1})
\end{equation}
\begin{equation}\label{fisher_06}
\overline{X}_{n}^{2}(\overline{X}_{n}-1)=\mu ^{2}(\mu -1)+ n^{-1/2}\mathbb{G}_{n}(L_{2}),
\end{equation}
\Bin where
\begin{equation*}
L_{2}=\mu^{2}h_{1}+(\mu -1)(2\mu h_{1})=\mu(3\mu-2)h_{1}.
\end{equation*}
\Bin So the denominator $\eqref{fisher_04}$/$\eqref{fisher_06}$ is expanded as
$$
denom=(2-\mu)\sigma^{2}-\mu^{2}(\mu-1)+n^{-1/2}\mathbb{G}_{n}(L_{3})+o_{\mathbb{P}(n^{-1/2})},
$$
\Bin where
$$
L_{3}=[\sigma^{2}+2\mu(2-\mu)-\mu(3\mu-2)]h_{1}+(2-\mu)h_{2}.
$$
\Bin Hence
\begin{eqnarray*}
\hat{a}&=&\frac{2\mu^{2}+n^{-1/2}\mathbb{G}_{n}(4\mu h_{1})+o_{\mathbb{a}(n^{-1/2})}}{Denom}\\
&=&\frac{2\mu^{2}}{(2-\mu)\sigma^{2}-\mu^{2}(\mu -1)+n^{-1/2}\mathbb{G}_{n}(H)+o_{\mathbb{a}(n^{-1/2})}},
\end{eqnarray*}
\Bin where
$$
H=\frac{2\mu}{(2-\mu)\sigma^{2}-\mu^{2}(\mu -1)}h_{1}-\frac{2\mu^{2}}{((2-\mu)\sigma^{2}-\mu^{2}(\mu -1))^{2}}L_{3}.
$$
\Bin Let us set
\begin{eqnarray*}
\alpha&=&(2-\mu)\sigma^{2}-\mu^{2}(\mu -1)\\
\beta&=&\sigma^{2}+2\mu(2-\mu)-\mu(3\mu-2).
\end{eqnarray*}
\Bin Then
$$
L_{3}=\beta h_{1}+(2-\mu)h_{2}.
$$
\Bin Hence
\begin{eqnarray*}
\notag H&=&(\frac{4}{\beta}\mu-\frac{2\mu^{2}}{\beta})h_{1}-\frac{2\mu^{2}(2-\mu)}{\beta^{2}}h_{2}\\
&=&\frac{2\mu(2-\mu)}{\beta}h_{1}-\frac{2\mu^{2}(2-\mu)}{\beta^{2}}h_{2}. \ \ \ \blacksquare
\end{eqnarray*}
\Bin
\section{Conclusions and perspectives} \label{sec5} Moment estimators for four statistical distributions been studied through their asymptotic Gaussian laws with the help of the \textit{fep} tool. Chi-square omnibus tests have been derived for each distribution. The results have been simulated and the chi-square tests revealed themselves efficient for small sample sizes. The R codes of the simulations are attached to the paper in an appendix. The main perspective is to develop a full chapter in with the study of a large number of distributions.
\textbf{Acknowledgment}. The authors Niang and Ngom express their thanks to Professor Lo for guidance, and moral and financial assistance.\\
\newpage
|
{
"timestamp": "2021-12-10T02:02:58",
"yymm": "2112",
"arxiv_id": "2112.04589",
"language": "en",
"url": "https://arxiv.org/abs/2112.04589"
}
|
\section{Introduction}
\label{s:intro}
Algebraic generating functions (GFs) occur throughout combinatorics and its applications. Combinatorial situations that frequently lead to algebraic GFs include: recursive structures described by context-free languages, such as binary trees and constrained (random) walks; RNA secondary structures; applications of the kernel method, the quadratic method for maps, and generalizations.
Our catalog of test problems \cite{GrMW2021} lists 20 algebraic GFs occurring in the recent research literature, 19 of which are multivariate.
A classic univariate example is the Catalan GF
$$
\frac{1 - \sqrt{1 - 4x}}{2x} = \sum_{n\geq 0} a_n x^n
$$
where $a_n$ can be interpreted as the number of noncrossing partitions of $\{1, \dots, n\}$, binary trees with $n$ nodes, rooted ordered trees with $n$ edges, Dyck paths of semilength $n$, etc. Asymptotics of coefficients of univariate algebraic GFs, such as this one, can be derived systematically via the local \emph{singularity analysis} of Flajolet and Odlyzko \cite{FlOd1990} combined with a global analysis of singularities of the GF \cite{FlSe2009, Chab2002}.
Thus the Catalan GF is easily shown to have $n$th coefficient asymptotic to $4^{n}/\sqrt{\pi n^3}$, which in this simple case can be verified using the explicit formula $a_n = (2n)!/((n+1)!n!)$ and Stirling's approximation.
However, in more than one variable, there is no such general procedure, and there are many situations where we genuinely want asymptotic information about specific coefficients of multivariate GFs. For example, a Schr\"{o}der tree is a rooted plane tree on $n$ leaves where each non-leaf vertex has at least two children. The GF encoding the number $a_{m, n}$ of Schr\"{o}der trees with $m$ leaves and $n$ vertices satisfies the algebraic equation
$$V(x, y) := \sum_{m, n = 0}^\infty a_{m, n} x^my^n = xy + \frac{y(V(x,y))^2}{1- V (x, y)}.$$
For large {$m$ and $n$}, approximately how many trees with {$m$ leaves and $n$ vertices} are there?
The most obvious method of attack on such a problem is to try to mimic the custom contour of integration used by Flajolet and Odlyzko. Apart from the PhD thesis work of Greenwood~\cite{Gree2018}, little has been done in this direction. For the results of that substantial work to apply, an explicit formula for the GF in the form $H^{-\beta}$ is required for some $\beta \not\in \mathbb{Z}_{< 0}$. The dominant singularity of $1/H$ must be a smooth minimal critical point, $H$ must be analytic near the origin, and $H$ must satisfy some partial derivative constraints at the critical point. Creating a general method from this seems rather daunting. {The full version of this paper will include examples where this method will not apply.}
Multivariate asymptotics are considerably more difficult to derive than in the univariate case, owing to the much greater range of geometries for the singular set of the GF. Even rational functions prove highly nontrivial, in sharp contrast to their algorithmic analysis in the univariate case. Previous work by Bender, Richmond and Gao \cite{BeRi1983, GaRi1992} deals with limit laws for multivariate situations including algebraic GFs, but does not address the detailed asymptotic questions we need here (corresponding in their model to estimating the probability generating function).
However, the long-running ACSV program \cite{PeWi2013}
has by now made rational multivariate GFs quite manageable in many cases. Our main approach here is to
embed the array of coefficients of an algebraic GF into a higher-dimensional array represented by a rational function, and then use the existing ACSV theory to derive asymptotics. This idea was suggested in \cite{RaWi2012a} but has barely been explored since then.
Here we make progress toward a systematic method, by classifying all examples in our recently collated test problem collection \cite{GrMW2021}, and presenting complete results in Section \ref{s:examples} for some of them. Complete details are in the accompanying Sage worksheet, \url{http://acsvproject.com/AlgSage.ipynb}. {To compute asymptotics for the GF $f(\mathbf{x})$:
\begin{enumerate}
\item {\bf Preprocess.} Adding finitely many terms to $f$ or making a variable substitution may help attain necessary embedding conditions, and may lead to nicer \emph{combinatorial} embeddings, where all coefficients are non-negative. (See Section \ref{ss:preprocess}.)
\item {\bf Embed.} Using Proposition \ref{pr:saf1}, encode the coefficients of $f(\mathbf{x})$ as the elementary diagonal of a rational function $F(Y, \mathbf{x})$ in one additional variable. This step requires that $f$ is divisible by some variable occurring in $f$ and that the minimal polynomial for $f$ has a non-zero derivative at the origin. (See Section \ref{ss:embed}.)
\item {\bf Identify critical points.} If $F(Y, \mathbf{x}) = G(Y, \mathbf{x})/H(Y, \mathbf{x})$, use a system of polynomial equations derived from $H$ to identify critical points that may contribute to the asymptotics. Because of space, we restrict to the case where we can find an embedding where $1/H(Y, \mathbf{x})$ is combinatorial, allowing us to identify minimal critical points more easily.
We leave more advanced ACSV issues, such as noncombinatorial embeddings and critical points at infinity, to follow-up work. (See Section \ref{ss:ACSV}.)
\item {\bf Compute asymptotics.} Once the contributing critical points are identified, asymptotics may be computed algorithmically. (See our Sage worksheet.)
\end{enumerate}
}
\section{Basic theory}
\label{s:theory}
Let $F(\xx) = \sum_{\rr} a_\rr \xx^\rr$ be a GF in $d$ variables that is algebraic with minimal polynomial $P(Y, \xx)$.
For the Catalan example above, $P(Y,x) = xY^2 - Y + 1$%
.
\subsection{Embedding using rational GFs}
\label{ss:embed}
\begin{pr}[\cite{Furs1967}]
\label{pr:furst}
Suppose that $f$ is a univariate algebraic power series defined by the polynomial $P(Y,x)$, that $f(0) = 0$, and that $\partial P/\partial Y(0,0) \neq 0$. Then $f$ is the diagonal $\Delta F$ of
the power series
$$
F(Y, x):= \frac{Y^2P_Y(Y,Yx)}{P(Y, Yx)}.
$$
\end{pr}
The condition that {$f(0) = 0$} is necessary for the formula, since the constant term of
$Y^2P_Y(Y,Yx)$ is zero. Also, $\partial P/\partial Y(0,0) \neq 0$ is necessary: it says that the algebraic function defined by $P$ has a single branch of multiplicity $1$ that passes through the origin.
\begin{eg}
\label{eg:catalan}
The Catalan GF has nonzero constant term, so there are two obvious ways to apply Proposition \ref{pr:furst}. Subtracting the constant term yields the embedding into
$$
F_1(Y,x) = \frac{Y\left(1 - 2Y^2x - 2Yx\right)}{1 - (Y^2x + 2Yx + x)}
$$
We could instead multiply by $x$ to obtain the shifted Catalan GF whose $n$th coefficient is $a_{n-1}$. This yields an embedding into
$$
F_2(Y,x) = \frac{Y(1-2Y)}{1-Y-x}.
$$
In other words, $a_n$ is the $(n,n)$ coefficient of $F_1$ and the $(n+1,n+1)$ coefficient of $F_2$.
\end{eg}
There is a substantial lack of uniqueness in the embedding, since adding any bivariate rational GF with zero diagonal will not change the diagonal.
Proposition \ref{pr:furst} in fact generalizes to a less well known result in an arbitrary number of variables, which also allows for a single branch of higher multiplicity. When $d\geq 3$ there are various notions of diagonal. {The \emph{elementary diagonal} of a GF $F(\mathbf{x})$ is the $(d-1)$-variate GF encoding the coefficients from $F$ where $x_1$ and $x_2$ have matching powers, $\sum_{\mathbf{r}} a_{r_2, r_2, r_3, \ldots, r_{d}} x_2^{r_2} x_3^{r_3} \cdots x_{d}^{r_{d}}$.}
\begin{pr}[\cite{Safo2000}, Lemma {2}]
\label{pr:saf1}
Suppose that $f$ is an algebraic power series given as a branch of
$P(f(\xx), \xx) = 0$, that $f$ is divisible by $x_1$ and that in some neighborhood of $\zero$, there is a factorization $P(Y, \xx) = (Y - f(\xx))^k u(Y, \xx)$ where $u(0,
\zero) \neq 0$ and $k \geq 1$ is an integer.
Then $f$ is the elementary diagonal of the rational function $F$ given by
$$
F(Y, \xx) =
\frac{Y^2 P_Y(Y,Yx_1, x_2, \dots, x_d)}{k P(Y,Yx_1, x_2, \dots, x_d)}.
$$
\end{pr}
\begin{eg}
\label{eg:narayana}
A bivariate refinement of the Catalan GF is the Narayana GF
$$
G(x,y):= \frac{1}{2x}{\left( 1 - x(y-1)
- \sqrt{1 - 2x(y + 1) + x^2(y -1)^2} \right)}
$$
which enumerates noncrossing partitions by set size and number of blocks, rooted ordered trees by edges and leaves, Dyck paths by semilength and number of peaks, etc. {The minimal polynomial $P(Y,x,y) = xY^2 - Y(1-x(y-1)) + 1$ specializes to the Catalan case on setting $y=1$.}
Now $G(x,y) - 1$ satisfies the hypotheses of Proposition \ref{pr:saf1} with $k=1$ (with respect to the variable $x$), and hence embeds into
$$
\frac{\left(1 - (2Y^2x + Yxy + Yx)\right)Y}{\left(1 - \left(Y^2x + Yxy + Yx + xy\right)\right)}.
$$
As expected, if we set $y = 1$, this GF specializes to $F_1$ in Example \ref{eg:catalan}. Similarly, if we instead multiply by $x$ before embedding, we obtain
$$
\frac{Y \left(1 - 2Y - Yx(y-1)\right)}{1 - x - Y - Y x(y-1)}
$$
which specializes to $F_2$ in Example \ref{eg:catalan}.
\end{eg}
More generally, we must separate the branches of the algebraic function, which requires more work, undertaken for example in \cite{Safo2000}. We intend to consider this case in a companion paper. For now, we concentrate only on cases where the hypotheses of Proposition \ref{pr:saf1} are met, which turns out to be almost all of the examples in \cite{GrMW2021}.
\subsection{Basics of ACSV}
\label{ss:ACSV}
We give a very brief overview of this (by now standard) material, and refer to \cite{PeWi2013} and \cite{Melc2021} for full details.
Given a rational $d$-variate GF in the form $F = G/H$ with a power series expansion at the origin, we may often derive asymptotics as follows. First represent the coefficient of $\xx^\rr := x_1^{r_1} \cdots x_d^{r_d}$ via the Cauchy Integral Formula with domain of integration a small torus (product of circles) centered at the origin. We then expand this torus via a homothety until we reach a \emph{minimal critical point} (a type of point with algorithmically checkable properties) lying on the boundary of the domain of convergence. Replacing by a local residue integral we may evaluate this using stationary phase methods. In the case where the minimal critical point {$\mathbf{w}$} is a smooth point of the variety $H = 0$, this leads directly to an asymptotic expansion {for $\mathbf{r} = n \cdot \hat{\mathbf{r}}$ with $\hat{\mathbf{r}} \in (0, \infty)^d$ fixed as $n \to \infty$}
$$
{\left[\mathbf{x}^{\mathbf{r}}\right] F(\mathbf{x}) \sim \mathbf{w}^{-\mathbf{r}}} \sum_{k\geq 0} a_k |\rr|^{(1-d)/2-k}
$$
where the coefficients can be determined algorithmically in terms of derivatives of $G$ and $H$, {and $\mathbf{w}$ and $a_k$ vary with $\hat{\mathbf{r}}$}. The simplest explicit formula in terms of $G$ and $H$ is only for $k = 0$, but the rational GFs we obtain via the embedding procedures above always have $a_0 = 0$ \cite{RaWi2012b}. Computer algebra implementations computing coefficients $a_k$ are built into Sage \cite{Raic}.
Another key observation is that if $1/H$ is \emph{combinatorial} (all its coefficients are nonnegative), then minimal singularities exist and for each minimal critical point $\ww$, the point $(|w_1|, \dots |w_d|)$ is also a minimal critical point. If the series is \emph{aperiodic} (the {subgroup} generated by the support is all of $\mathbb{Z}^d$) then there is a unique minimal critical point. In particular, if $H = 1 - K$ where $K$ is an aperiodic polynomial with nonnegative coefficients, which is the case in many of our examples below, for each direction there is a unique minimal critical point supplying asymptotics in that direction, and it is strictly minimal and lies in the positive orthant.
In the general noncombinatorial case, it can be hard to determine the contributing points. Minimal points are still desirable because the contour shifting implicit in the above description is still guaranteed to work in the noncombinatorial case. However minimal points need not exist, and algorithmically determining them is considerably harder. Much recent work has been done, using Morse theory, to deal with this case. In this article we discuss only {examples that can be manipulated until a combinatorial embedding is found}, leaving the asymptotic analysis for {noncombinatorial cases for} a future article.
\section{Worked examples}
\label{s:examples}
We categorize the examples in \cite{GrMW2021} according to the type of embedding obtained. We present a single illustrative example in each category in full detail, and give less detail in other cases. Full computations are given in the accompanying Sage worksheet.
\subsection{Notes on preprocessing}
\label{ss:preprocess}
Assuming that we have an algebraic power series $f$, there are two main hypotheses needed before we can embed using Proposition \ref{pr:saf1}. They are: (H1) $f$ is divisible by some variable occurring in $f$ and (H2) $\partial P/\partial Y$ does not vanish at the origin.
If $f$ does not vanish at the origin, then H1 cannot apply. In the univariate case, we can remedy this by subtracting the constant term or multiplying by the variable, as we showed above for the Catalan example. More generally, for the first approach we can subtract initial terms of the power series expansion of $f$. The following result gives a simple necessary condition for this to work. {Let $\xx_j^\circ = (x_1, \ldots, x_{j - 1}, 0, x_{j + 1}, \ldots, x_d)$.}
\begin{pr}
\label{pr:necessary}
{If $f - f_0$ satisfies H1 with respect to the variable {$x_j$}, for some polynomial $f_0$, then {$f(\xx_j^\circ)$} is a polynomial.}
\end{pr}
\begin{proof}
If we substitute {$x_j=0$}, then we need {$f(\xx_j^\circ) - f_0(\xx_j^\circ)$} to be identically zero.
\end{proof}
Note that if we already satisfy H1 but still apply such an \emph{additive substitution}, then the truth value of H2 will not change. To see this, write $\tilde{f} = f - f_0$ and denote its minimal polynomial by $\tilde{P}$. Then $\partial \tilde{P}/\partial Y$ evaluated at the origin is equal to $\partial P/\partial Y$ evaluated at $\xx = 0, Y = f_0(\zero) = 0$. Thus in this case such a substitution is unnecessary if our goal is simply to embed. However as we see in Example \ref{eg:schroeder}, the nature of the rational function that we embed into may change, and this may be useful.
In all dimensions, the second method of \emph{multiplicative substitution} results in something satisfying H1. The Catalan example shows that we may also satisfy H2. However the situation seems quite tricky:
the ternary tree analog of the shifted Catalan GF is defined by $Y^3 - xY + x^2 = 0$, and this does not satisfy H2. We do not pursue this approach further in the present article, for space reasons and because it seems less promising.
There is a third trick we can use. Assuming we already have $f(\zero) = 0$ but H1 is not satisfied, we can make a \emph{monomial substitution}. For example, in two variables we may have $(x,z) \mapsto (x,xz)$. This ensures that every term is now divisible by $x$, so H1 holds for the transformed GF $\tilde{f}(x,z) = f(x,xz)$. Furthermore there is no change in the truth value of H2, since the minimal polynomial of $\tilde{f}$ is $P(Y, x, xz)$ and this does not change the value of $\partial P /\partial Y$ evaluated at the origin. We do need to keep more careful track of the relation between the indices
in the original power series and the rational function.
\subsection{Embedding is combinatorial after additive substitution}
\label{ss:additive}
In 6 of our 20 catalogued examples, we can immediately embed into a combinatorial GF once we remove the constant term, and in 4 examples a similar approach with more terms works. In addition to the examples shown below, the catalog examples \#8 (Eu; refinement of noncrossing partitions), \#9 (Do\v{s}lic et al.; RNA secondary structures), \#13 (Flajolet and Sedgewick; patterns in trees), \#14 (the Narayana numbers), \#16 (new and old leaves), and \#19 (Bousquet-M\'{e}lou \& Rechnitzer; bar graphs) lie in this category.
\subsubsection{A univariate algebraic family \cite[Example 4]{GrMW2021}}
\label{eg:callan}
As a warm-up we deal with a univariate problem.
In \cite{Call2007}, Callan studied the family
\[
G(x): = G_{a,b}(x) = \frac{1 - ax - \sqrt{1-2ax + (a^2 - 4b)x^2}}{2bx^2},
\]
for constants $a$ and $b$.
Let $f(x) = G(x) - 1$, and observe that $f$ satisfies the minimal polynomial
\[
P({Y, x}) = x^2b Y^2 + (2bx^2 + ax - 1)Y + (x^2b + ax) .
\]
Since $P_Y(0, 0) \neq 0$, we can use Proposition \ref{pr:saf1} to embed $f$ in the GF
\[
F({Y, x}) = \frac{Y^2 P_Y({Y, xY})}{P({Y, xY})}
= \frac{{\left(1 - \left(2 \, Y^{3} b x^{2} + 2 \, Y^{2} b x^{2} + Y a x\right)\right)} Y}{1 - \left(Y^{3} b x^{2} + 2 \, Y^{2} b x^{2} + Y b x^{2} + Y a x + a x\right)}.
\]
We look at the coefficients of $F$ in the direction $\rr = [1, 1]$ because $[x^n] f(x) = [{Y^nx^n}] F({Y, x})$. Let $H({Y, x})$ be the denominator of $F({Y, x})$. Then, critical points satisfy the system of equations
$\left\{H = 0,\ \ xH_x - YH_Y = 0\right\}$.
When $b \neq a^2, a^2/4$, this system yields two points:
\[
{(Y_1, x_1)} = \left(\frac{-\sqrt{b}}{(a - \sqrt{b})(a - 2\sqrt{b})}, -\frac{a \sqrt{b} - b}{b}\right),
{(Y_2, x_2)} = \left(\frac{\sqrt{b}}{(a + \sqrt{b})(a + 2\sqrt{b})}, \frac{a \sqrt{b} + b}{b}\right).
\]
All critical points are smooth because the system of equations $\{H = 0,\ \ xH_x - YH_Y = 0, H_x = 0\}$ has no solutions.
The coordinates of {$(Y_2, x_2)$} are always positive for any choice of $a, b > 0$, and this point is strictly minimal.
We now use, for example, \cite[Thm 5.2]{Melc2021} to find an asymptotic expansion for the coefficients of $F({Y, x})$.
We end up with
\begin{equation} \label{eq:Example1-4-Asymptotics}
[{Y^nx^n}] F({Y, x}) = \frac{b^{-3/4}}{2\sqrt{\pi}} n^{-3/2} (a + 2\sqrt{b})^{n + 3/2} + O(n^{-5/2}).
\end{equation}
Note that when $b = a^2$ or $b = a^2/4$, the non-dominant critical point is no longer relevant, but a similar analysis yields the same asymptotic formula.
We compare this to the asymptotic expansions derived using \cite[Cor. 2]{FlOd1990}. The original GF $G(x)$ has a unique algebraic singularity at $x = \frac{1}{a + 2\sqrt{b}}$ for $a, b > 0$. Additionally, if $\alpha_1 = \frac{1}{a + 2\sqrt{b}}$ and $\alpha_2 = \frac{1}{a - 2\sqrt{b}}$, then as $x \to \alpha_1$,
\[
\sqrt{1-2ax + (a^2 - 4b)x^2} \sim \sqrt{\alpha_1} \left(1 - \frac{x}{\alpha_1}\right)^{1/2} \sqrt{(\alpha_2 - \alpha_1)(a^2 - 4b)}.
\]
This implies that as $n \to \infty$
\[
[x^n] \sqrt{1-2ax + (a^2 - 4b)x^2} = -\frac{b^{1/4}}{\sqrt{\pi}}n^{-3/2} (a + 2\sqrt{b})^{n - 1/2} + O(n^{-5/2}).
\]
Plugging this expansion into the equation for $G(x)$ gives a matching asymptotic expression to Equation \eqref{eq:Example1-4-Asymptotics}.
\subsubsection{Dissections \cite[Example 10]{GrMW2021}}
\label{eg:dissections}
Drmota \cite[p.376]{Drmo2009} enumerates dissections of polygons (where the polygons have a marked edge) using a bivariate GF $A(x,y)$, where $x$ counts the number of vertices in the polygon, and $y$ counts the total number of edges in the dissection.
He gives the following minimal polynomial for $A$:
\[
A(x, y) = xy^2(1 + A(x, y))^2 + xy(1 + A(x, y)) \cdot A(x, y).
\]
Since $A$ is divisible by $x$, using Proposition \ref{pr:saf1} immediately we embed into
$$F({Y, x, y}) = \frac{{\left(1 - \left(2 \, Y^{2} x y^{2} + 2 \, Y^{2} x y + 2 \, Y x y^{2} + Y x y\right) \right)} Y}{1 - \left(Y^{2} x y^{2} + Y^{2} x y + 2 \, Y x y^{2} + Y x y + x y^{2}\right)}.
$$
We note that $[x^{pn} y^{(1-p)n}]A(x, y) = [{Y^{pn}x^{pn}y^{(1-p)n}}] F({Y, x, y})$, and find that there is a single smooth critical point in this direction:
\[
{(Y, x, y)} = {\left(\frac{1 - 2 \, p}{p}, \frac{p(3p-1)^2}{(1-2p)^3}, \frac{(1-2p)^2}{(3p-1)(1-p)}\right)}.
\]
This critical point has positive coordinates when $1/3 < p < 1/2$. This corresponds to the range of $p$-values where $A(x, y)$ has positive coefficients%
. It is clear from the denominator of $F$ that in this case, the critical point is strictly minimal.
Thus
for $1/3 < p < 1/2$:
\[
[x^{pn} y^{(1-p)n}]A(x, y) = \frac{\sqrt{1-p}}{2\pi p^2 \sqrt{3p - 1}} \cdot \frac{1}{n^2} \cdot \left(\frac{(1-p)^{1-p}}{(1-2p)^{2-4p}(3p-1)^{3p-1}}\right)^n + O\left(\frac{1}{n^3}\right).
\]
\if01
\subsubsection{Non-crossing partitions with $k$ visible blocks}
\label{eg:Eu}
Let $b_{n, k}$ be the number of circular non-singleton non-crossing partitions of $[n]$ with $k$ visible blocks. Eu presents \cite[Prop. 63, p.107]{Eu2002}:
\[
B(x, u) := \sum_{n, k \geq 0} b_{n, k} u^k x^n = \frac{1 + x - \sqrt{1 - 2x + x^2 - 4x^2u}}{2x(1 + xu)}.
\]
Using Proposition \ref{pr:saf1}, $B(x, u) - 1$ can be embedded in a GF $F(x, u, Y)$ where $x$ is attached to $Y$. We search for asymptotics of $[x^{np} u^{n(1-p)}] B(x, u)$. For circular representations of partitions, every block is visible. Thus, for a partition of $[n]$ with no singleton blocks, there are between $1$ and $\lfloor n/2\rfloor$ blocks. This corresponds to $p \in (2/3, 1)$.
In the $(x, u, Y)$-direction $\rr = (p, (1-p), p)$, there is a single smooth strictly minimal critical point with positive coordinates that leads to the asymptotic expansion
\[
[x^{np}u^{n(1-p)}] B(x, u) = \frac{\sqrt{p}}{2\pi (2p - 1)^2 \sqrt{3p - 2}} \cdot \frac{1}{n^2} \cdot \left(\frac{(p-1)^{2p - 2} p^p}{(3p-2)^{3p-2}} \right)^n + O\left(\frac{1}{n^3}\right).
\]
\fi
\subsubsection{Assembly trees \cite[Example 15]{GrMW2021}}
\label{eg:assembly}
B\'{o}na and Vince
\cite{BoVi2013} define the concept of \emph{assembly
tree} of a graph and show that the (exponential) GF for the number of
assembly trees of the complete bipartite graph $K_{rs}$ is
$$
f(x,y) = \sum_{rs}a_{rs}x^ry^s = 1 - \sqrt{(1-x)^2 + (1-y)^2 - 1}.
$$
Note that $f(0,y) = y$ and $f(x,0) = x$; indeed, the first order Maclaurin expansion is $x+y$.
We replace $f$ by $\tilde{f}:=f - x - y$, which obviously has no effect on asymptotics of coefficients, and we can embed $\tilde{f}$ via Proposition \ref{pr:saf1} into
$$
F({Y, x, y}) = \frac{{\left(1 - \left(Y x + Y + y \right)\right)} Y}{1 - \left(Y x + x y + Y/2 + y\right)}.
$$
Here, we analyse the direction $\rr = [{p, p, (1-p)}]$ for variables $({Y, x, y})$ and $p \in (0, 1)$, and obtain a pair of smooth critical points, only one of which is strictly minimal with positive coordinates.
Let $\xi := 1 + 4p - 4p^2$. This leads to the asymptotic formula
\[
\frac{\sqrt{(1 + \xi)\sqrt{\xi} - 2\xi}\left(1 + \xi^{-1/2}\right)}{4\pi p(1-p)} \cdot \frac{1}{n^2} \cdot
\left( \frac{(\sqrt{\xi} - 2p + 3)}{4(1-p)^2} \cdot \left( \frac{1 + (1-2p)\sqrt{\xi}}{4p^2}\right)^p \right)^n + O\left(\frac{1}{n^3}\right).
\]
For example, when $p = 1/2$ we are looking at the diagonal case $K_{rr}$, and obtain
\[
[x^ny^n]f(x, y) = \frac{1}{\pi 2^{9/4}} \cdot \frac{1}{n^2} \cdot (6 + 4\sqrt{2})^n + O\left(\frac{1}{n^3}\right).
\]
This agrees with \cite[Thm 4.11]{BoVi2013} in the exponential rate, but not in the constant --- ours is correct, as we confirmed by numerical checks.
\subsubsection{Schr\"{o}der trees by leaves and vertices \cite[Example 17]{GrMW2021}}
\label{eg:schroeder}
Recall this class from the introduction. Using Proposition \ref{pr:saf1} on the minimal polynomial for $V$ (with respect to the variable $y$) will yield a noncombinatorial embedding. Instead, we use the embedding $F({Y, x, y})$ for $V(x, y) - xy$.
Checking in the $({Y, x, y})$ direction $\rr = ({p, p, (1-p)})$ for $p \in (1/3, 1/2)$ (corresponding to non-zero coefficients) yields a single strictly minimal critical point with positive coordinates.
Then, we obtain
\[
[x^{np}y^{n(1-p)}]V(x, y) = \frac{\sqrt{3p - 1}}{2\pi p^2\sqrt{1-p}} \cdot \frac{1}{n^2} \cdot \left(\frac{(1-2p)^{4p-2}}{(1-p)^{p-1}(3p-1)^{3p-1}} \right)^n + O\left(\frac{1}{n^3}\right).
\]
\subsection{Embedding is combinatorial after a monomial substitution}
\label{ss:melczer trick}
When the trick of subtracting a polynomial from $f$ does not work, we can move to another level.
In 2 of our 20 examples, a monomial substitution is effective.
\subsubsection{Bivariate generalization of Catalan numbers \cite[Example 7]{GrMW2021}}
\label{sss:Cossali}
The next example comes from \cite{Coss2003}.
Let $$g(n, m) = \frac{(2n + m)!}{m!n!(n + 1)!}.$$
with corresponding GF
\[
L(x, z) := \sum_{q = 0}^\infty \sum_{m = 0}^\infty g(m, q)x^m z^q = \frac{(1 - z) - \sqrt{(1 - z)^2 - 4x}}{2x}.
\]
Then $L(0,0) \neq 0$ and Proposition \ref{pr:necessary} rules out simple additive substitutions, since $L(x,0)$ is the Catalan GF and L'H\^{o}pital's rule shows that $L(0,z)$ is also not a polynomial. Instead we change variables by replacing $z$ by $xz$. Then $L(x, xz) - 1$ satisfies
\[
P = Y^{2} x + Y x z + 2 \, Y x + x z - Y + x,
\]
which leads to the embedding
\[
F({Y, x, z}) = \frac{1 - \left(2 Y^{2} x + Y x z + 2 Y x\right) Y}{1 - \left(Y^{2} x + Y x z + 2 Y x + x z + x\right)},
\]
where $[x^nz^k] L(x, z) = [{Y^{n + k}x^{n+k} z^n}] F({Y, x, z})$. Rescaling so $n+k$ becomes $n$, we consider $[x^{pn} z^{(1-p)n}] L(x, z)$ for a constant $p \in (0, 1)$ as $n \to \infty$. This corresponds to analysing $[{Y^n x^n z^{(1-p)n}}]F({Y, x, z})$, giving us the direction $\rr = ({1, 1, (1-p)})$.
There is a single smooth strictly minimal critical point with positive coordinates in this direction, yielding
\[
[x^{pn} z^{(1-p)n}] L(x, z) = \frac{\sqrt{1 - p^2}}{2\pi (1-p)p^2} \cdot \frac{1}{n^2} \cdot \left( \frac{1 + p}{1-p} \cdot \frac{(1-p^2)^p}{p^{2p}}\right)^n + O\left(\frac{1}{n^3}\right).
\]
\subsubsection{Bicolored Motzkin Paths \cite[Example 18]{GrMW2021}}
In \cite[Lemma 2.1]{Eliz2021}, Elizalde derives the GF for bicolored Motzkin paths. Here, such a path starts and ends at the $x$-axis, never passes below the $x$-axis, and takes steps $U = (1, 1), D = (1, -1)$, and two (colored) types of horizontal steps, $H_1 = (1, 0)$ and $H_2 = (1. 0)$. Let $a_{m, n}$ be the number of such paths with $m$ total $U$ or $H_1$ steps and $n$ total $D$ or $H_2$ steps. Then
\[
M(x, y) := \sum_{m, n = 0}^\infty a_{m, n} x^m y^n = \frac{1 - x - y - \sqrt{(1 - x- y)^2 - 4xy}}{2xy}.
\]
Using Proposition \ref{pr:saf1} on $M(x, xy) - 1$ gives the asymptotic formula
\[
[x^{pn} y^{(1-p)n}] M(x, y) = \frac{1}{2\pi (1-p)^2p^2} \cdot \frac{1}{n^2} \cdot \left( \frac{(1-p)^{2p - 2}}{p^{2p}}\right)^n + O\left(\frac{1}{n^3}\right).
\]
\if01
\subsubsection{General quadratic GFs}
Suppose that $F(\xx)$ is algebraic with minimal polynomial of the form $P(Y,\xx) = a(\xx) Y^2 + b(\xx)Y + c(\xx)$. A necessary condition for the hypotheses of
Proposition \ref{pr:saf1} to hold is $b(\zero) \neq 0$.\edit{check}.
\fi
\if01
\subsubsection{Leaves in trees}
Recall the Catalan and Narayana examples from the introduction.
There is a further refinement \cite{ChDE2006} that considers different
types of leaves in a rooted ordered tree.
Call a leaf of such a tree \emph{old} if it is the leftmost
child of its parent, and \emph{young} otherwise. They enumerate such trees
according to the number of old leaves, number of young leaves and number of
edges, finding the algebraic equation
$$
G(x,y,z) = 1 + \frac{z(G(x,y,z) - 1 + x)}{1 - z(G(x,y,z) - 1 + y)}.
$$
Thus $G$ is defined by the polynomial
$$
P(Y,x,y,z) = ***
$$
and we obtain the embedding
$$
***
$$
\fi
\if01
\subsection{More difficult examples}
So far we have dealt with *** out of *** examples in \cite{GrMW2021}. Of the remaining ones in the catalog \cite{GrMW2021}, Proposition \ref{pr:saf1} applies for almost all \edit{how many}, but we have not yet found a combinatorial embedding in those examples. For example, bilateral Schr\"{o}der paths (see \url{https://oeis.org/A063007}) from $(0,0)$ to $(n,n)$ having $k$ North steps are enumerated by
$$G(t, z) = \frac{1}{\sqrt{1-2z-4tz+z^2}.}$$
The embedding $F({Y, t, z})$ found by using Proposition \ref{pr:saf1} and attaching $z$ to $Y$ yields no affine critical points in any relevant direction $\rr = [{(1-p), p, (1-p)}]$ for $p \in (0, 1)$.
For only *** of them does H2 fail to hold. Such cases can be treated by
Safonov's algorithm \cite{Safo2000}, which we intend to do in work currently in preparation.
\fi
\section{Further discussion}
\label{s:discuss}
We believe we have shown that our strategy for computing asymptotics of multivariate algebraic series is a good one.
Further developments in ACSV theory, some already near completion, will allow analysis of noncombinatorial embeddings. We intend to take this up in a future work, and some cases will be challenging. For example, bilateral Schr\"{o}der paths
from $(0,0)$ to $(n,n)$ having $k$ North steps are enumerated by
$$G(t, z) = \frac{1}{\sqrt{1-2z-4tz+z^2}.}$$
The embedding $F({Y, t, z})$ found by using Proposition \ref{pr:saf1} and attaching $z$ to $Y$ yields no affine critical points in any relevant direction.
We note further that Furstenberg and Safonov are not the only relevant authors for embedding procedures. For example, Denef \& Lipshitz \cite{DeLi1987} give a somewhat more complicated method, which when applied to the example $x/\sqrt{1-x}$ yields, instead of the rather difficult to analyse embedding given by Proposition \ref{pr:furst}, the simple GF $2xY/(1-x-Y)$. This indicates a brighter future for the general program of computing multvariate asymptotics by reducing the
algebraic to the rational case.
\section{Acknowledgements}
This material is based upon work supported by the National Science Foundation under Grant Numbers 1641020 and 1916439, an NSERC Discovery Grant, an NSERC USRA and the American Mathematical Society Mathematics Research Community, Combinatorial Applications of Computational Geometry and Algebraic Topology.
\bibliographystyle{plain}
|
{
"timestamp": "2022-06-29T02:05:12",
"yymm": "2112",
"arxiv_id": "2112.04601",
"language": "en",
"url": "https://arxiv.org/abs/2112.04601"
}
|
\section{Introduction}
Probe vehicles or Lagrangian sensors (\cite{herrera2010incorporation}) can be considered as tracking-device equipped vehicles that can report critical information such as direct travel time, speed (\cite{ramezani2012estimation, jenelius2013travel,jenelius2015probe,hans2015applying,zheng2018traffic}) flow (\cite{duret2017traffic,seo2019fundamental}) or inferred delay (\cite{florin2020towards}) and queue lengths (\cite{bae2019spatio, wang2020queue}) as they traverse transportation networks. Commercial taxis, volunteers, transit buses, maintenance vehicles, commercial trucks, etc., can report their location and timestamps through cellphones and GPS devices for improved traffic operations or better planning. Traffic state can be estimated using Collected data. The accuracy of these estimates depends on the accuracy of reported sensor data and the penetration of the number of data received from the vehicles. Regardless, observing mobile data from transportation networks gives critical coverage for dynamic traffic behavior. This study presents a method for estimating queue length given that (1) probe vehicles can be observed on a lane accurately and infer the order of vehicles in a queue, (2) we can deduce the beginning of queue start time to probe vehicle arrival times (e.g., relative to the beginning of red duration at a signal), and (3) we can track the number of probe vehicles in the queue. Assuming these data are available, using the combinatorics approach, we develop queue length estimators that can be used for any queues without requiring primary arrival rate or probe vehicle market penetration rate (or percentage) parameters.
Researchers have extensively studied the queue length estimation problem by proposing parametric (\cite{zhao2019various, zhao2021maximum, zhao2021hidden}) and nonparametric methods. In this paper, we focused on the review of nonparametric ones. Jin and Ma presented a study on a nonparametric Bayesian method for traffic state estimation (\cite{jin2019non}). In their study, they developed a generalized modeling framework for estimating traffic states at signalized intersections. The framework is nonparametric and data-driven, and no explicit traffic flow modeling is required.
Wong et al. estimated the market penetration rate (probe proportion or percentage) (\cite{wong2019estimation}). Based on probe vehicle data alone, they proposed a simple, analytical, nonparametric, and unbiased approach to estimate penetration rate.
The method fuses two estimation methods. One is from probabilistic estimation and the second from samples of probe vehicles which is not affected by arrival patterns. It uses PVs and all vehicles ahead of the last PV in the queue.
Gao et al. presented queue length estimations (QLEs) based on shockwaves and backpropagation neural network (NN) sensing (\cite{gao2019connected}). The approach uses PV data and queue formation dynamics. It uses the shockwave velocity to predict the queue length of the non-probe vehicles. The NN is trained with historical PV data. The queue lengths at the intersection are obtained by combining the shockwave and NN-based estimates by variable weight. Tan et al. introduced License Plate Recognition (LPR) data in their study to fuse with the vehicle trajectory data, and then developed a lane-based queue length estimation method (\cite{tan2020fuzing}). Authors matched the LPR with probe vehicle data. They obtained the probability density function of the discharge headway and the stop-line crossing time of vehicles. They presented the lane-based queue lengths and overflow queues. Wang et al. proposed a QLE method on street networks using occupancy data (\cite{wang2013modeling}). Their key idea is using the speed decrease as the queue increase downstream of loop detector. This would result in higher occupancy at constant volume-to-capacity ratios.
Using VISSM simulation, they generated data for various link length, lane number, and bus ratio.
They fit a logistic model for the queue length and occupancy relationship. Then, queue lengths were estimated using multiple regression models.
The purpose of this paper is to model queue lengths at intersections generally without assuming random arrivals or any primary parameters or estimating parameters.
Unlike fundamental non-parametric queue length estimations from arrival and service distributions (\cite{schweer2015nonparametric, goldenshluger2016nonparametric, goldenshluger2019nonparametric,singh2021estimation}), our method uses mathematical techniques from combinatorics to derive discrete conditional probability mass functions of observed information about the queue and derive moments of the distributions without depending on arrival or service distributions. The approach presented in this study essentially extends the results from (\cite{Comert2009, Comert2013}) where \cite{Comert2009} presented a conditional probability mass function for probe location information and \cite{Comert2013} provided closed-form queue length estimators given probe vehicle location and time information for Poisson arrivals.
The paper is organized as follows: In section \ref{intro}, the approach is defined to set up derivations. In section \ref{PEV}, we use combinatorial arguments and present a closed form of the sum of the probabilities in Eqs.~(\ref{probability-1}) and (\ref{probability-2}). The result obtained in section \ref{PEV} enables us to define a probability mass function. We show that this probability distribution is Negative Hypergeometric. We use the results for the mean and variance of the distribution to derive formulas for the queue length estimators. In section \ref{field}, we present numerical examples for the behavior of the derived estimators and show the performance of the estimators using field data.
We summarize our findings and discuss possible future research directions in section \ref{sctconc}.
\section{Problem Definition}
\label{intro}
Probe vehicles (PVs) and partially observed systems through inexpensive sensors are facilitating real-time queue length estimations.
The essential information needed for the estimation is primary parameters such as flow rate and percent of probe vehicles. However, both parameters are dynamic. Especially in real-time applications, like cycle-to-cycle or shorter-term queue lengths at signalized intersections, one would need to collect data for a few cycles to estimate these parameters.
The parameters can then be updated and used in such applications. Assuming random arrivals, in (\cite{comert2016queue,comert2020cycle}), it is shown that
at least 10 cycles of CV data would be needed to start using queue length estimators.
Our goal in this paper is to model queue lengths at intersections without assuming random arrivals or any primary parameters or estimating parameters. The problem is intuitive. The conditional probability of the location (order) of the last probe vehicle can be calculated by $P(L=l|M=m,N=n)=\binom{l-1}{m-1}/\binom{n}{m}$ given number of probe vehicles and the total number of vehicles in the queue. In this probability mass function, $L$ is the location of the last probe vehicle, $M$ is the number of probe vehicles in the queue, and $N$ is the total queue (\cite{comert2009}). We can see that this does not assume any arrival pattern or parameter and only depends on probe vehicle data.
Certainly, it is not taking advantage of queue joining time $T$ of CVs with respect to signal timing. Also, we know that signal timing can be considered an integer (or increments of 5-10 seconds). There is also the physical constraint of the vehicle following, which can be 1-2 seconds $s$ even if vehicles arrive in a platoon.
\begin{figure}[ht]
\centering
\includegraphics[width=0.6\linewidth]{intersection8.png}
\caption{Example queue with queue length, PV data, and arrival-no arrival spots}
\label{fig_1}
\end{figure}
The problem is described on a single lane of approach and can equivalently be expressed as an urn modeling (see Fig.~\ref{fig_1}). Consider the approach lane queue formed in $2R$ (i.e., $R$ is red phase) time intervals where in one time interval there can be at most one arrival. So, in this set up there can be at most one arrival per $0.5$ seconds. This can be thought of as the minimum possible time gap and can be updated in the formulations derived. In Fig.~\ref{fig_1}, we have $l$ arrivals in $2t$ time intervals among these $2t-1$ contain $m-1$ CVs, $2R-2t$ has $n-l$ arrivals. Now, the problem is a negative inference, meaning $n$ is changing as in Negative Binomial, so, we are interested in $P(N=n|L=l,M=m,T=t,R)$, i.e., probability of having $N=n$ arrivals within $R$ time interval given $L=l, T=t, M=m$. Calculating this probability, we obtain Eq.~(\ref{probability-1}) or equivalently Eq.~(\ref{probability-2}).
\begin{equation}
\label{probability-1}
\frac{\binom{2t}{l-m} \binom{2R-2t}{n-l}}{\binom{2R}{n-m}}
\end{equation}
\begin{equation}
\label{probability-2}
\frac{\binom{n-m}{l-m} \binom{2R-(n-m)}{2t-(l-m)}}{\binom{2R}{2t}}
\end{equation}
We can then calculate expected values to get the mean ($E(n|l,t,m,R)$ or the queue length estimator) and the variance ($V(n|l,t,m,R)$) of the estimator. However, we first need to
\begin{enumerate}[i.]
\item verify if this is a valid probability mass function.
\item find the normalizing denominator for a valid probability mass function.
\item simplify to forms that can be used as input-output models like $E(n|l,t,m,R)=l+(1-p)\lambda (R-t)$ in (\cite{comert2013simple}).
\item show if this
approach leads to one of the known negative probability mass functions (e.g., Negative Hypergeometric). This could facilitate (iii).
\end{enumerate}
\section{Probability Mass Function, Expected Value, and Variance}
\label{PEV}
We first provide combinatorial arguments and derive a closed-form for the sum of the function in Eq. (\ref{probability-2}). \\
\begin{theorem}
\label{theorem1}
Let $\ell$, $t$, $R$, and $m$ be as defined in the preceding section. Then
\begin{equation}
\sum_{n=\ell}^{2R-2t+\ell}\frac{\binom{n-m}{\ell-m}\binom{2R+m-n}{2t+m-\ell}}{\binom{2R}{2t}}
=\frac{2R+1}{2t+1}\\
\end{equation}
or equivalently
\begin{equation}
\sum_{n=\ell}^{2R-2t+\ell}\binom{n-m}{\ell-m}\binom{2R+m-n}{2t+m-\ell}=\binom{2R+1}{2t+1}
\end{equation}
\end{theorem}
\bf Proof. \rm Observe $\binom{n-m}{\ell-m}=\binom{n-m}{n-\ell}$ and $\binom{2R+m-n}{2t+m-\ell}=\binom{2R+m-n}{2R-2t-n+\ell}$ and replace $n'=n-\ell$ so that Eq.~(4) becomes
\begin{eqnarray}
\label{step2}
\sum_{n'=0}^{2R-2t}\binom{n'+\ell-m}{n'}\binom{2R+m-n'-\ell}{2R-2t-n'}=\binom{2R+1}{2t+1}
\end{eqnarray}
Make another re-indexing $\ell'=\ell-m$ and hence Eq. (\ref{step2}) takes the form
\begin{eqnarray}
\label{step3}
\sum_{n'=0}^{2R-2t}\binom{n'+\ell'}{n'}\binom{2R-\ell'-n'}{2R-2t-n'}=\binom{2R+1}{2t+1}
\end{eqnarray}
The \it "negativization" \rm (reminiscent of the Euler's \it gamma reflection formula \rm $\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)}$) of binomial coefficients $\binom{-a+b}b=(-1)^b\binom{a-1}b$ allows to convert
$\binom{\ell'+n'}{n'}=(-1)^{n'}\binom{-\ell'-1}{n'}$ and $\binom{2R-\ell'-n'}{2R-2t-n'}=\binom{2t-\ell'+2R-2t-n'}{2R-2t-n'}=(-1)^{2R-2t-n'}\binom{\ell'-2t-1}{2R-2t-n'}$.
Therefore,
\begin{eqnarray}
\label{step4}
\sum_{n'=0}^{2R-2t}\binom{n'+\ell'}{n'}\binom{2R-\ell'-n'}{2R-2t-n'}\nonumber \\
=\sum_{n'=0}^{2R-2t}\binom{-\ell'-1}{n'}\binom{\ell'-2t-1}{2R-2t-n'}
\end{eqnarray}
The well-known \it Vandermonde-Chu \rm identity states $\sum_{k=0}^y\binom{x}k\binom{z}{y-k}=\binom{x+z}y$. Applying this to Eq. (\ref{step4}) and engaging $\binom{-a+b}b=(-1)^b\binom{a-1}b$
(one more time) yields
$$\sum_{n'=0}^{2R-2t}\binom{-\ell'-1}{n'}\binom{\ell'-2t-1}{2R-2t-n'}=$$ $$\binom{-2t-2}{2R-2t}=(-1)^{2R-2t}\binom{2R+1}{2R-2t}=\binom{2R+1}{2t+1}$$
The proof is complete. $\square$
\bf Remark. \rm The identity just proved shows that Eq. (3) or Eq. (4) is independent of the parameters $m$ and $\ell$.
\newline
We can see that the identity proved in the above theorem enables us to revise Eq.~(\ref{probability-2}) and define a probability mass function. We can divide both sides of the identity by the expression on the right-hand side of the identity to get one on the right-hand side (i.e., the right-hand side of the identity is the normalizer of the probability distribution, which we explain below). Note that additional results from combinatorics and discussions are presented in the Appendix.
In Negative Hypergeometric distribution (\cite{johnson2005univariate}), the probability of having $k$ successes up to the $r^{th}$ failure given sample size of $S$ and maximum possible queued vehicles $K$ is given by
\begin{equation}
\label{probneghyp}
p(k| r, K, S) = \frac{\binom{k+r-1}{k}\binom{S-r-k}{K-k}}{\binom{S}{K}}
\end{equation}
where $S$ is the sample size (time capacity for arrivals and non-arrivals), $K$ is the total number of successes (arrivals) in $S$, $r$ is the number of failures (non-arrivals), and $k$ is the number of successes (realizations of arrivals). The probabilities sum to $1$. For the Negative Hypergeometric distribution, the expected value $E(k|r,K,S)$ and the variance $V(k|r,K,S)$ are given by Eqs.~(\ref{expvalneghyp}) and (\ref{varneghyp}).
\begin{equation}
\label{expvalneghyp}
E(k|r,K,S) = \frac{rK}{S-K+1}
\end{equation}
\begin{equation}
\label{varneghyp}
V(k|r,K,S) =\frac{rK(S+1)(S-K-r+1)}{(S-K+1)^2(S-K+2)}
\end{equation}
In the probability mass function of the Negative Hypergeometric distribution (Eq.~(\ref{probneghyp})), let $S = 2R + 1$, $K = 2R - 2t$,
$r = l - m + 1$, and $k = n - l$. Then the result proved in the theorem above gives the following probability mass function, which is a Negative Hypergeometric distribution since $\sum_{n=l}^{2R-2t+l} \binom{n - m}{l - m}\binom{2R + m - n}{2t + m - l}= \binom{2R + 1}{2t + 1}$. Notice that with these assignments arrivals and non-arrivals are fixed and probability of the total queue $N=n$ is calculated with known $l,m,t,R$.
\begin{equation}
\label{probmassfn}
p(N=n|l,m,t,R) = \frac{\binom{n - m}{l - m}\binom{2R + m - n}{2t + m - l}}{\binom{2R + 1}{2t + 1}}
\end{equation}
From the formulas for the expected value and variance of the Negative Hypergeometric distribution, we get the following formulas for the expected value (Eq.~(\ref{expvalneghyp})) and variance (Eq. (\ref{varneghyp})) of this probability distribution in Eq.~(\ref{probmassfn}). Note that $L=l,M=m,T=t,R$ are basic information from CVs, not primary parameters (arrival or penetration rate of probe vehicle in the traffic stream). We also do not require steady-state behavior if this Probe vehicle information is available. The expected queue length and its variance are short-term ($R$ seconds or time interval) estimators.
\begin{figure}[h!]
\captionsetup{aboveskip=3pt,belowskip=-5pt}
\captionsetup[subfigure]{aboveskip=2pt,belowskip=-1pt}
\centering
\begin{subfigure}{0.49\columnwidth}
\centering
\includegraphics[width=0.99\columnwidth]{p22.png}
\caption{$p(N|T=20,L,M=2,R=45)$ in Eq.~(\ref{probmassfn})}
\label{fig_2a}
\end{subfigure}
\captionsetup[subfigure]{aboveskip=2pt,belowskip=-1pt}
\begin{subfigure}{0.49\columnwidth}
\centering
\includegraphics[width=0.99\columnwidth]{p2.png}
\caption{$p(N|L,M=2,R=45)$ in Eq.~(\ref{probmassfn2})}
\label{fig_2b}
\end{subfigure}
\caption{Example behavior of conditional probabilities}
\label{fig_2}
\end{figure}
The expected value $E(n|l,t,m,R)$ can be determined by
\begin{eqnarray*}
E(N|l,m,t,R)&=& \sum_{n=l}^{2R-2t+l}{\frac{n(2R+1)}{(2t+1)}\frac{\binom{n-m}{l-m} \binom{2R+m-n}{2t+m-l}}{\binom{2R+1}{2t+1}}}\\
&=& \sum_{n'=0}^{2R-2t}{\frac{n'(2R+1)}{(2t+1)}\frac{\binom{n'+l'}{n'} \binom{2R-l'-n'}{2R-2t-n'}}{\binom{2R+1}{2t+1}}}
\end{eqnarray*}
where $n'=n-l$, $l'=l-m$, and $\frac{(2R+1)}{(2t+1)}$ is the normalizer.
By Eqs. (\ref{expvalneghyp}) and (\ref{varneghyp}), simplified expected value or the queue length estimation 1 and the variance can be obtained as in Eqs.~(\ref{eqn_nhg1}) and (\ref{eqn_vnhg1}).
\begin{eqnarray}
\label{eqn_nhg1}
E(N_1=n_1|l,m,t,R) &= & l+\frac{(l-m+1)(2R-2t)}{2t+2}\nonumber \\
&=& l+\frac{(l-m+1)(R-t)}{t+1}
\end{eqnarray}
\begin{eqnarray}
\label{eqn_vnhg1}
V(N_1=n_1|l,m,t,R)=&\frac{(l-m+1)(2R+2)(2R-2t)}{(2t+2)(2t+3)}
[1-\frac{l-m+1}{2t+2}]
\end{eqnarray}
Alternatively, from Eq.~(\ref{probmassfn2}), we can get the following
equivalent estimator without CV time ($T$) information (Eq.~(\ref{eqn_nhg2})) and its variance in (Eq.~(\ref{eqn_vnhg2})).
\begin{equation}
\label{probmassfn2}
p(N=n|l, m, C) = \frac{\binom{C-n+m}{C-n} \binom{n-m}{n-l}}{\binom{C}{l}}
\end{equation}
where $C$ is capacity or maximum possible arrivals, $J=C-2l$, $r=l-m+1$, $K=C-l$, and $k=n-l$. Note that, with time discretization, we can infer $t$ from $l$.
The expected value $E(n|l,m,R)$ is given by
\begin{eqnarray*}
E(N=n|l,m,R) &=& \sum_{n=l}^{C+l}{\frac{n(l+1)}{(C+1)}\frac{\binom{C-n+m}{C-n} \binom{n-m}{n-l}}{\binom{C}{l}}} \\
&=& \sum_{n'=0}^{C}{\frac{n'(l+1)}{(C+1)}\frac{\binom{n'+l'}{n'} \binom{C-l'+n'}{C-l'-n'}}{\binom{C}{n'}}}
\end{eqnarray*}
where $n'=n-l$, $l'=l-m$, and $\frac{(l+1)}{(C+1)}$ is the normalizer for valid probability mass function.
\begin{equation}
\label{eqn_nhg2}
E(N_2=n_2|l,m,C)=l+\frac{(l-m+1)(C-l)}{l+2}
\end{equation}
\begin{equation}
\label{eqn_vnhg2}
V(N_2=n_2|l,m,C) =\frac{(l-m+1)(C+2)(C-l)}{(l+2)(l+3)}[1-\frac{l-m+1}{l+2}]
\end{equation}
One of the advantages of the derived estimators in Eqs.~(\ref{eqn_nhg1}) and (\ref{eqn_nhg2}) is that the denominators are nonzero since $L\geq0$. This enables us to estimate queues even if there is no probe vehicle in the queue. The behavior of conditional probabilities, expected values, and variances are shown in Figs.~\ref{fig_2}-\ref{fig_4}. We can see in Fig.~\ref{fig_2} that the likelihoods are right to the $N=l$ values. In Fig.~\ref{fig_3}, as queue time joining of the last probe vehicle increases, then the expected queue length gets closer to $L=l$ for both models. Similarly, in Fig.~\ref{fig_4}, the variance of the estimated queue length reduces as $l$ and $t$ increase. Having time information also shows smoother behavior compared to having only location information.
\begin{figure}[h!]
\captionsetup{aboveskip=3pt,belowskip=-5pt}
\captionsetup[subfigure]{aboveskip=2pt,belowskip=-1pt}
\centering
\begin{subfigure}{0.49\columnwidth}
\centering
\includegraphics[width=0.99\columnwidth]{e12.png}
\caption{$E(N|T,L,M=2,R=45)$ in Eq.~(\ref{eqn_nhg1})}
\label{fig_3a}
\end{subfigure}
\captionsetup[subfigure]{aboveskip=2pt,belowskip=-1pt}
\begin{subfigure}{0.49\columnwidth}
\centering
\includegraphics[width=0.99\columnwidth]{e2.png}
\caption{$E(N|L,M,R=45)$ in Eq.~(\ref{eqn_nhg2})}
\label{fig_3b}
\end{subfigure}
\caption{Example behavior of conditional expectations}
\label{fig_3}
\end{figure}
\begin{figure}[h!]
\captionsetup{aboveskip=3pt,belowskip=-5pt}
\captionsetup[subfigure]{aboveskip=2pt,belowskip=-1pt}
\centering
\begin{subfigure}{0.49\columnwidth}
\centering
\includegraphics[width=0.99\columnwidth]{v1.png}
\caption{$V(N|T,L,M=2,R=45)$ in Eq.~(\ref{eqn_nhg1})}
\label{fig_4a}
\end{subfigure}
\captionsetup[subfigure]{aboveskip=2pt,belowskip=-1pt}
\begin{subfigure}{0.49\columnwidth}
\centering
\includegraphics[width=0.99\columnwidth]{v2.png}
\caption{$V(N|L,M,R=45)$ in Eq.~(\ref{eqn_vnhg2})}
\label{fig_4b}
\end{subfigure}
\caption{Example behavior of conditional variances}
\label{fig_4}
\end{figure}
\section{Evaluation with Field Queue Length Data}
\label{field}
To show the effectiveness of the estimators developed,
we used 2014 ITS World Congress Connected Vehicle Demonstration Data (\cite{dataset}). Authors' previous works used this field data for evaluating range sensor inclusion and filtering for queue length estimation (\cite{comert2020cycle,comert2021queue}). Results for this study are new. For completeness, assumptions and set up are reported again. The dataset contains manually collected queue lengths at the intersection of Larned and Shelby streets in Detroit, Michigan, between September $8$ and $10$, $2014$. The number of observations per day are $98$, $254$, and $135$, respectively. During data collection, probe vehicles were identified with the blue $X$s. Each row of data includes hour, minute, second of an observation, the maximum queue lengths, and the number of probe vehicles in these queues (i.e., $M$ in the formulations above) from left, center, and right lanes of Larned street approach.
The dataset provides $M=m$ and $C$ cycle time values but not the information of $L$ and $T$
from PVs. Hence, we generated random variates of this information from Uniform ($L=l$ location $l\sim U(m,n)$) and Gamma ($T=t$ queue joining time $t\sim\mathcal{G}a(l,\frac{C}{2n})$) distributions for all lanes independently and repeated for $1000$ random seeds. Note that, integer values are used for $L,M$, and $T$. Overall average of estimation errors are reported to compare models. In addition, followings are assumed related to the traffic signal and the dataset:
\begin{enumerate}
\item Back of queue observations are obtained at the end of red phases (vary cycle-by-cycle). The time between two observations is assumed to be the cycle length ($C$) and red phases are assumed to be half ($R=C/2$).
\item There is no steady growth of queue and many zero queue values. Thus, the overflow queues are omitted . The data was collected during low to medium $\rho$ (i.e., volume-to-capacity ratio$=0.50$). Regardless, $\rho$s are also calculated using estimated arrival rates.
\item The capacity of the approach was approximated by the observed overall maximum queue value of $10$ vehicles within $70$ seconds ($10\times3600/35$=$1029$ vehicles per hour or $0.286$ vehicles per second ($vps$) saturation flow rate). These values are used essentially in the Highway Capacity Manual (HCM) from a manual and back of queue calculations. Note that the values may not be reflecting actual capacity and phase splits;
however, we compare and report against true queue lengths. This would provide insights into the accuracy of our approach.
\end{enumerate}
Compared HCM delay (i.e., $Delay$) and back of queue (i.e., $Q {back}$) models are given in Eqs. (\ref{eqn_hcm}) and (\ref{eqn_qb}). These models are approximations for given time intervals (e.g., $15$ minutes) and fully observed traffic.
\begin{eqnarray}
\label{eqn_hcm}
d_1=\frac{C}{2}[\frac{(1-G/C)^2}{1-[min(1,\hat{X})G/C]}]\nonumber\\
d_2=900T[(\hat{X}-1)+\sqrt{(\hat{X}-1)^2+\frac{8kI\hat{X}}{cT}}]
\end{eqnarray}
where $d$=$d_1\times PF+d_2+d_3$ is control delay seconds per vehicle, $d_2$ is uniform delay, $PF$ is progression factor due to arrival types, $d_2$ is random delay component, and $d_3$ is delay due to initial queue. In this study, only $d_1+d_2$ are considered with $d_3=0$ since no overflow queue is assumed. $PF=1.0$ is used for random arrivals. Volume-to-capacity is $\hat{X}=\hat{\rho}=\frac{\hat{\lambda}}{0.286}$. Green time $G$ is in seconds $s$, $C$ is cycle time in $s$. $T$ is the analysis period in hours where in cycle-to-cycle estimations $T_i=C_i/3600$ is assumed where $i$ denotes cycle number. $k$ is incremental delay factor, and $0.5$ is assumed for fixed time like movement. $I=1$ upstream filtering is assumed for no interaction with nearby intersections, and capacity is $c=1029$ $vph$. Note that in our calculations, uniform delay is the main component updated by changing $G$ and $C$ values. Queue lengths are approximated by Little's formula $d\lambda$ where $d$ and $\lambda$ are both calculated at each cycle using $M$ number of probe vehicles in the queue. This method is based on HCM 2000 (\cite{prassas2020highway, ni2020signalized}).
Another estimation approach adopted from (\cite{kyte2014operation}) that is used to calculate cycle-to-cycle back of queues (see Eq.~(\ref{eqn_qb})).
\begin{eqnarray}
\label{eqn_qb}
Q_{back}=\hat{v}(R+g_s)
\end{eqnarray}
where, $Q_{back}$ is back of the queue in vehicles, $v=\lambda$ is arrival rate in vehicles per second ($vps$), $R$ is the red duration in seconds $s$, and $g_s$ is queue service time that is calculated $\hat{v}R/(x-\hat{v})$ with $x$ is the saturation flow rate (i.e., assumed to be $0.286$ vps). All the values $R$, $g_s$, and $\hat{v}$ except $x$ are changing cycle-to-cycle.
Alternative estimators from \cite{comert2016queue} are denoted by Est.1 and Est.2 in Eqs.~(\ref{est.1}) and (\ref{est.2}), respectively. These queue length estimators are in the form of $E(n|l,m,t,R)=l+(1-\hat{p})\hat{\lambda}(R-t)$ with two different primary parameter estimator combinations: $\{\hat{\lambda}_1=\frac{l}{R}, \hat{p}_1=\frac{m}{l}\}$ and $\{\hat{\lambda}_2=\frac{(l-m)}{t}+\frac{m}{R}, \hat{p}_2=\frac{mt}{mt+(l-m)R}\}$.
\begin{equation}
\label{est.1}
E(N_1|l,m,t,R)=I(m>0)[l+(l-m)(1-\frac{t}{R})]+I(m=0)[(1-\frac{\bar{m}}{\bar{l}})(\bar{l}+(\bar{l}-\bar{m})(1-\frac{\bar{t}}{R}))]
\end{equation}
\begin{eqnarray}
\label{est.2}
E(N_2|l,m,t,R)=I(m>0)[m+\frac{R(l-m)}{t}]+I(m=0)[(1-\frac{\bar{m}\bar{t}}{\bar{m}\bar{t}+(\bar{l}-\bar{m})R})(\bar{m}+\frac{R(\bar{l}-\bar{m})}{\bar{t}})] \nonumber \\
\centering
=I(m>0)[m+\frac{(l-m)R}{t}]+I(m=0)[\bar{m}+\frac{(\bar{l}-\bar{m})R}{\bar{t}}]
\end{eqnarray}
where $I(.)$ is the indicator function. When there is no probe vehicle in the queue (i.e., $I(m=0)$), we use the average of previous probe vehicles' information as we need to estimate arrival rate ($\lambda$) and probe percentage ($p$). Notation $M_{1:i}$ represents values from cycle $1$ to $i$ and $\bar{m}_{1:i}=\sum_{j=1}^{i}{\frac{m_j}{i}}$, $\bar{l}_{1:i}=\sum_{j=1}^{i}{\frac{l_j}{i}}$, and $\bar{t}_{1:i}=\sum_{j=1}^{i}{\frac{t_j}{i}}$. Average error values are given in Table~\ref{tab_data} for $T\sim\mathcal{G}a(l,\frac{C}{2n})$. Fig.~\ref{fig_box} is given to demonstrate if assumed interarrivals are impacting the accuracy of the estimators.
\begin{table}[h!]
\centering
\caption{\textcolor{black}{Estimation results with RMSE errors in [vehs/cycle] with $T\sim\mathcal{G}a(l,$C$/(2n))$}}
\label{tab_data}
\scalebox{0.85}{
\begin{tabular}{c c c c c c c cc}
\hline\noalign{\smallskip}
& Lane& Avg. $p$ & Est.1 & Est.2 & NP.Est.1 & NP.Est.2 & Delay & Q back \\
\hline\noalign{\smallskip}
\multirow{3}{*}{Sep08}
& L & 13\% &1.09 &1.01 & 1.01 & 1.01 & 1.37 & 1.25 \\
& C & 21\% &0.72 &0.78 &0.69 &0.70 &1.33 &1.26 \\
& R & 7\% &0.56 &0.55 &0.60 &0.60 &0.66 &0.64 \\
\hline
\multirow{3}{*}{Sep09}
& L & 10\% &1.22 &1.05 &0.99 &0.99 &1.38 &1.13 \\
& C & 26\% &1.14 &0.96 &1.09 &1.09 &1.81 &1.38 \\
& R & 2\% &0.34 &0.35 &0.54 &0.54 &0.39 &0.41 \\
\hline
\multirow{3}{*}{Sep10}
& L & 7\% &2.68 &2.43 &2.48 &2.48 &2.81 &2.52 \\
& C & 18\% &1.48 &1.32 &1.73 &1.73 &2.28 &1.63 \\
& R & 1\% &0.84 &0.77 &0.77 &0.77 &1.06 &1.26 \\
\noalign{\smallskip}\hline\noalign{\smallskip}
\end{tabular}
}
\end{table}
In Table~\ref{tab_data}, a summary of average queue length ($QL$) estimation errors in the root mean square is provided
(RMSE=$\sqrt{\sum_{i=1}^{n}{\frac{(QL_i-\hat{QL}_i)^2}{n}}}$). Average $p$ values are calculated from $\sum_{i=1}^{n}{\frac{m}{nQL_i}}$ for each lane. Since true maximum queues are not known, $p$ and $\lambda$ are estimated. HCM's control delay-based model and back of queue are denoted by $HCM_d$ and $Q_{back}$, respectively. The accuracy of the estimators is reported when probe vehicles are present in the queue ($p$=$\{10\%, 13\%, 18\%, 21\%$, $26\%\}$).
Example performances with $21\%$ penetration rates is given in Fig.~\ref{fig_est}. When there are probe vehicles in the queue, we can see that the proposed methods can follow the true maximum queue lengths closely. In Fig.~\ref{fig_box}, boxplots for overall errors are given. We can see that the model with new estimators provides slightly lower errors. However, errors are lower than delay-based $HCM_d$ and $Q_{back}$ methods. Our methods can estimate more accurately compared to $Q_{back}$.
\begin{figure}[h!]
\captionsetup{aboveskip=3pt,belowskip=-5pt}
\captionsetup[subfigure]{aboveskip=2pt,belowskip=-1pt}
\centering
\begin{subfigure}{0.49\columnwidth}
\centering
\includegraphics[width=0.99\columnwidth]{estimation.png}
\caption{Estimation on Sep 08 on center-lane}
\label{fig_est}
\end{subfigure}
\captionsetup[subfigure]{aboveskip=2pt,belowskip=-1pt}
\begin{subfigure}{0.49\columnwidth}
\centering
\includegraphics[width=0.99\columnwidth]{box.png}
\caption{Box plots for all estimation errors}
\label{fig_box}
\end{subfigure}
\caption{Performance of the proposed estimators $NP.Est.1$ and $NP.Est.2$}
\label{fig_5}
\end{figure}
\section{Conclusions}
\label{sctconc}
In this study, we derived two new nonparametric queue length estimation models for traffic signal-induced queues. The estimators only depend on signal phasing and timing information. The derivations involved fundamental experiment setup, and
the resulting
estimators were simple algebraic expressions. We did not assume independent arrivals at the intersection. The only assumption we made was discrete time intervals which are
reasonable as signal timing involves whole seconds; in fact, multiples of five seconds.
For independent approach lanes at traffic intersections, it is shown that conditional queue lengths given probe vehicle location, count, time, and analysis interval can be represented by a Negative Hypergeometric distribution.
The performance of the estimators derived
was compared with
parametric and simple highway capacity manual methods that use field test data involving probe vehicles. The results obtained from the comparisons show that the nonparametric models presented in this paper match the accuracy of parametric models.
The methods developed do not assume random arrivals of vehicles at the intersection or any primary parameters or involve parameter estimations.
In this study, we developed methods to estimate queue length at intersection approaches from probe vehicles. These probe vehicles could be traditional probe vehicles or connected vehicles that generate basic safety messages. Future research could study the models presented in this paper to a more complex intersection and a series of adjacent intersections with large traffic demand volume at these intersections.
\vspace{-10pt}
\section*{Acknowledgments}
This study is partially supported by the Center for Connected Multimodal Mobility ($C^{2}M^{2}$) (USDOT Tier 1 University Transportation Center) headquartered at Clemson University, Clemson, SC. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of $C^{2}M^{2}$ and the official policy or position of the USDOT/OST-R, or any State or other entity, and the U.S. Government assumes no liability for the contents or use thereof. It is also partially supported by U.S. Department of Homeland Security SRT Follow-On grant, Department of Energy-National Nuclear Security Administration (NNSA) PuMP, MSIPP IAM-EMPOWEREd, MSIPP, Department of Education MSEIP programs, NASA ULI (University of South Carolina-Lead), and NSF Grant Nos. 1719501, 1954532, and 2131080.
\vspace{-5pt}
\section*{Appendix}
\label{append}
\noindent
The results in Eq. (\ref{theorem1}) can be extended to the short sum runs from $n=\ell$ through $n=2R-2t$.
\smallskip
\begin{theorem}
\label{theorem2}
Let $\ell$, $m$, $n$, $R$, and $t$ be as defined in Theorem \ref{theorem1}. Then \it there is a recurrence formula for
$$\sum_{n=\ell}^{2R-2t}\binom{n-m}{\ell-m}\binom{2R+m-n}{2t+m-\ell}. \quad (5)$$ \rm
\end{theorem}
\bf Proof. \rm Denote the sum by $f(\ell)$ and the summand by $F(\ell,n)$ (after suppressing the remaining variables). Introduce the function
$G(\ell,n)=-\binom{n-m}{\ell+1-m}\binom{2R+m-n+1}{2t+m-\ell}$. Then, it is routine to verify that
$$F(\ell+1,n)-F(\ell,n)=G(\ell,n+1)-G(\ell,n). \quad (6)$$
Sum both sides of (6) for $n=\ell+1$ to $n=2R-2t$ (and telescoping on the right-hand side) to obtain
$$f(\ell+1)-f(\ell)+F(\ell,\ell)=G(\ell,2R-2t+1)-G(\ell,\ell+1).$$
Based on $F(\ell,\ell)=\binom{2R+m-\ell}{2t+m-\ell}, G(\ell,2R-2t+1)=-\binom{2R-2t-m+1}{\ell-m+1}\binom{2t+m}{\ell}$ and $G(\ell,\ell+1)=-\binom{2R+m-\ell}{2t+m-\ell}$, we infer the
\bf recursive relation \rm
$$f(\ell+1)-f(\ell)=-\binom{2R-2t-m+1}{\ell-m+1}\binom{2t+m}{\ell}. \qquad \square$$
\bf Corollary. \it From Theorem \ref{theorem2}, we get the following identity
$$\sum_{\ell=0}^{2R-2t}\binom{2R-2t-m+1}{\ell-m+1}\binom{2t+m}{\ell}=\binom{2R+1}{2t+1}. \quad (7)$$ \rm
\bf Proof. \rm This follows from the recurrence relation proved in Theorem \ref{theorem2} and the identity proved in Theorem \ref{theorem1}. $\square$
\bigskip
\noindent
\begin{theorem}
\label{theorem3}
The identity in (7) can be re-indexed and formulated as follows:
$$\sum_{m=\ell}^{R-t+\ell } {m\choose \ell}{R-m\choose t-\ell}={R+1\choose t+1}.$$
\end{theorem}
\bf Proof. \rm We offer a combinatorial argument.
Given natural numbers $\ell\le t\le R$, and $m$, we may consider the class of those $(t+1)$-subsets $\{x_0<x_1<\dots<x_t\}$ of $\{0,1,\dots,R\}$ such that $x_\ell=m$: these are exactly ${m\choose \ell}{R-m\choose t-\ell}$ (indeed the $\ell$ elements $x_0,\dots, x_{\ell-1}$ can be chosen freely into $\{0,\dots, m-1\}$, and so can the $t-\ell$ elements $x_{\ell+1},\dots,x_t$ into $\{m+1,\dots,R\}$. These classes, for $ \ell\le m\le R-t+\ell $ form a partition of all $(t+1)$-subsets of $[R+1]$, whence the sum of their cardinality is independent of $\ell$ and the identity.
$\square$
\bigskip
\noindent
\bf Remark. \rm The discrepancy in having a closed form and no closed form can be understood as follows: we know that $\sum_{k=0}^n\binom{n}k=2^n$, however there is no "nice evaluation" for $\sum_{k=0}^m\binom{n}k$ unless $m=n$. The bottom line is the former is summed over the full compact support of $\binom{n}k$ (in the sense, $\binom{n}k=0$ if $k<0$ or $k>n$. A similar analogy can be drawn with having the closed form $\int_{\Bbb{R}}e^{-x^2}dx=\sqrt{\pi}$ but nothing similar is available if the limit are altered to be any smaller subset than the full range $\Bbb{R}$, except for $[0,\infty)$.
|
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"yymm": "2112",
"arxiv_id": "2112.04551",
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"url": "https://arxiv.org/abs/2112.04551"
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"\\section{Introduction}\n\\label{intro}\nGraph-structured data have become ubiquitous in the real w(...TRUNCATED)
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"\\section*{ABSTRACT}\nParcel sorting operations in logistics enterprises aim to achieve a high thro(...TRUNCATED)
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