text string | source string |
|---|---|
3.880 7.838 6.384 0.910 0.940 0.970 ⌊4T/5⌋5.840 4.610 9.305 7.148 0.840 0.910 0.970 ⌊T/3⌋ 3.950 3.780 6.706 5.951 0.900 0.900 0.950 ϕ=−0.3⌊T/2⌋ 3.700 3.280 5.628 5.259 0.900 0.940 0.980 ⌊2T/3⌋2.780 2.530 4.324 3.711 0.920 0.940 0.990 ⌊4T/5⌋4.310 3.700 6.520 5.810 0.840 0.870 0.960 ⌊T/3⌋ 2.370 2.190 3.632 3.384 0.940 0.... | https://arxiv.org/abs/2503.23211v2 |
confidence intervals for the EEG data set. The true value of t= 85 is displayed as a point in all intervals. 88 E.2 Surveillance video data set We estimate the change points for applications related to the EEG data set and the surveil- lance video data set described in the manuscript using some alternative methods. The... | https://arxiv.org/abs/2503.23211v2 |
lobby (so-called action 1)” and “ second person walks into the lobby (so- called action 2)” using STD method. Note that the true value is 116 for action 1 and 174 for action 2. Action Pixel Estimated change points based on STD method 1 700 {84,48,109,37,65,91,120,102,161,132} 702 {79,12,104,90,129,109} 731 {91,109,96,1... | https://arxiv.org/abs/2503.23211v2 |
MODELING MAXIMUM DRAWDOWN RECORDS WITH PIECEWISE DETERMINISTIC MARKOV PROCESSE IN CAPITAL MARKETS Rolando Rubilar-Torrealba∗Lisandro Fermin†Soledad Torres‡ April 1, 2025 ABSTRACT We propose to model the records of the maximum Drawdown in capital markets by means a Piecewise Deterministic Markov Process (PDMP). We deriv... | https://arxiv.org/abs/2503.23221v1 |
, Instituto de Ingeniería Matemática, Universidad de Valparaíso, Chile; Universidad Central de Chile, 8370242 Santiago, Chile . Email: soledad.torres@uv.clarXiv:2503.23221v1 [q-fin.RM] 29 Mar 2025 APREPRINT - APRIL 1, 2025 that evolve over time and to understand the asymptotic behaviour of certain phenomena, where reco... | https://arxiv.org/abs/2503.23221v1 |
interested in the process for each time instant over the records that it can reach at the maximum Drawdown . For this stochastic process we consider the following definition of the occurrence times of a records in the process, such that Ti=inf{t >0 :D(t)> D(Ti−1)∧ ∃s > t, D (s)−D(t) = 0}, (2) withT0= 0andi∈N, where Tir... | https://arxiv.org/abs/2503.23221v1 |
Here, Jnis the state take by νton[Tn, Tn+1[. The matrix generator A= (aij)i,j∈Kof the process (νt)t∈R+is given by aij= λiqij for i ̸=j −λi(1−qii)for i =j. (4) The generator Ais stable and conservative; i.e.P j∈Kaij= 0andaij≥0fori̸=j. which implies that the solutions of the system are driven by this differential operat... | https://arxiv.org/abs/2503.23221v1 |
(5)with initial value R0∈[0,1[satisfy Rt=R0+ (1−R0)R0 t, (6) where the process (R0 t)t∈R+given by R0 t=X k≥1ρJk k−1Y i=1(1−ρJi)! 1 l(t≥Tk), (7) is the records process obtained when the initial condition is zero, R0 0= 0. Proof: Rtis given by equation (5) where ∆k=ρJk(1−rk)are given by the following recursive equations:... | https://arxiv.org/abs/2503.23221v1 |
ν0=j] =E[r+ρ−ρr+R0 t−rR0 t−ρR0 t+ρrR0 t|R0=r+ρ(1−r), ν0=j] =E[r+ (1−r)R0 t−ρ(r+ (1 + r)R0 t−1)|R0=r+ρ(1−r), ν0=j] =ρ+ (1−ρ)E[Rt|R0=r, ν0=j] =ρ+ (1−ρ)m(t, r, j). (13) Then, from (12) and (13) we have ∂m ∂t(t, r, ν ) = λνX j∈Kqνj(µj+ (1−µj)m(t, r, j)−m(t, r, ν )). This differential equations system can be written as a sy... | https://arxiv.org/abs/2503.23221v1 |
defined in (19) for the two-state case with initial value r= 0. It is observed that the mean values increase rapidly at the beginning and then decrease with time, approaching the limit 1. The figure is like the learning curve, which describes a situation in which the task may be easy to learn and the learning progressi... | https://arxiv.org/abs/2503.23221v1 |
Rtis given by V ar(Rt) = a+b c+r−1 (c−µ)ect+ r2−a+b c−r−1 (c−µ)1 ect− 1−1−r eµν0t2 =r−1 (c−µ)ect+ r2−1−r−1 (c−µ)1 ect+ 21−r eµν0t−1−r eµν0t2 ≤2(1−r) eµν0t, (31) providing the functional form of the variance of process Rtfor the case of one-state. Analysing the function that characterises the second moment of th... | https://arxiv.org/abs/2503.23221v1 |
]), finance (see, [Liu et al.(2012)Liu, Margaritis and Wang]). First, we must characterise the changes of state of the process Rtfor each of the jumps. Each jump of the process is characterised by a time occurrence and a jump size. Both characteristics are modelled independently by defining a transition matrix Q. The a... | https://arxiv.org/abs/2503.23221v1 |
calculated for those process parameters Rtis placed in the figure. We can see that the computational approximation of the process is very similar to the one obtained through the analytical solution. This approach makes it possible to characterise the evolution of the main statistics of the Rtprocess numerically, provid... | https://arxiv.org/abs/2503.23221v1 |
2, repeating the process until it converges. The following Algorithm 3 allows to generate the 14 APREPRINT - APRIL 1, 2025 classification of the state labels. The first cycle generates the new classification of the labels, given the maximum likelihood calculations defined in Algorithm 2. The second cycle allows the est... | https://arxiv.org/abs/2503.23221v1 |
obtained according to Algorithms 2 and 3 we can numerically calculate the mean and the variance of the estimated processes which we will call ˆRr. In figures 7 and 8 we show the estimation of the mean and variance of the S&P500 - Rtprocess. 16 APREPRINT - APRIL 1, 2025 Figure 7: Sample path of ˆRtprocess, 10,000 simula... | https://arxiv.org/abs/2503.23221v1 |
Business Media. [Fermín and Lévy-Véhel(2020)] Fermín, L.J., Lévy-Véhel, J., 2020. Variability and singularity arising from a piecewise-deterministic markov process applied to model poor patient compliance in the multi-iv case. Jour- nal of Applied Statistics 47, 2525–2545. URL: https://doi.org/10.1080/02664763.2019.171... | https://arxiv.org/abs/2503.23221v1 |
Bayesian Inference for High-dimensional Time Series with a Directed Acyclic Graphical Structure Arkaprava Roy, Anindya Roy and Subhashis Ghosal University of Florida, University of Maryland Baltimore County, and North Carolina State University Abstract Inmultivariatetimeseriesanalysis, understandingtheunderlyingcausalr... | https://arxiv.org/abs/2503.23563v4 |
the opportunity for developing computationally efficient DAGestimationmethodsinapenalizedframeworkusingaugmentedLagrangianoptimizationwhere additional penalties can be included to encourage sparsity and other structural properties. Several approaches have been proposed for a fully Bayesian DAG estimation under known [A... | https://arxiv.org/abs/2503.23563v4 |
< j. If the process is stationary, the DAG structure stays invariant over time. Such a causal form of dependence in a multivariate ‘stationary’ time series (Yt:t= 0,1, . . .)may be encapsulated through a linear transform, modeling the ‘causal residual process’ Yt−WY tseparately from the graph structure. The operator Wd... | https://arxiv.org/abs/2503.23563v4 |
approximation, specification of prior distributions, and posterior sampling strategies are described in Section 3. The convergence properties of the posterior distribution are studied in Section 4. Extensive simulation studies to compare the performance of the proposed Bayesian method with other possible methods are ca... | https://arxiv.org/abs/2503.23563v4 |
with ‘zero’ diagonal. The last two terms in the expressions help impose the DAG-ness restriction (c.f., Zheng et al. [2018]), and the adaptive LASSO [Zou, 2006] penalty in the second term forces sparsity in the DAG structure as in Xu et al. [2022]. While more choices proposed in Yu et al. [2019], Bello et al. [2022] ma... | https://arxiv.org/abs/2503.23563v4 |
= 1, . . . , S, Xk,t=BUZ k,t, k = 1, . . . , p, Xk,t= (X(s) k,t:s= 1, . . . , S ), where Zk,t= (Zs,j,t:s= 1, . . . , S )are independent univariate stationary time-series with unit variance having spectral densities fs,j(ω),Uis an orthogonal matrix and Bis the spherical coor- dinate representation of Cholesky factorizat... | https://arxiv.org/abs/2503.23563v4 |
Diag (a)refers to the diagonal matrix with entries ain that order. 7 3.2 Prior distribution We put the commonly adopted nonparametric Bayesian prior given by a finite random series for the spectral densities. We choose the standard B-spline basis to form the series because of their shape and order-preserving properties... | https://arxiv.org/abs/2503.23563v4 |
. , p, r= 1,2, . . ., control local shrinkage of the elements in ξj,r, whereas τrcontrols column shrinkage of the rth column. –Next, let ηr= (η1,r, . . . , η J,r)∼NK(0, σκP−1), σ ξ, σκ∼IG(c1, c1); here Pis the second-order difference matrix to impose smoothness given by P=QTQ, where Qis theK×(K+ 2)matrix of the second ... | https://arxiv.org/abs/2503.23563v4 |
modeling, we need to impose the following two restrictions in the supports of D,W, and the spectral density parameters as in Theorem 3 of Roy et al. [2024]. Conditions on prior: (P1) Diagonal entries d1, . . . , d pand the range of the functions f1, . . . , f plie in a fixed, compact subinterval of (0,∞). (P2) Eigenval... | https://arxiv.org/abs/2503.23563v4 |
as 2 and 4. This leads to different levels of sparsity in W. The weights for the edges are generated from Unif ((−2,−0.5)∪(0.5,2)). The entries in Dare generated as absolute values of N (7,2). The S×Scorrelation matrix Bis generated in two steps. First, a precision matrix is generated using g-Wishart where the underlyi... | https://arxiv.org/abs/2503.23563v4 |
0.02 0.05 0.10 0.10 48 0.01 0.02 0.02 0.04 0.08 0.08 Table 4: MCC when Zℓ’s are generated following a Gaussian process with a cosine covariance kernel. Time points Expected neighbors = 2 DAG-OUT A-NOTEARS LINGAM PC rank-PC 32 0.81 0.46 0.53 0.27 0.29 48 0.83 0.50 0.54 0.42 0.37 Expected neighbors = 4 DAG-OUT A-NOTEARS ... | https://arxiv.org/abs/2503.23563v4 |
and Rental and Leasing 5 Professional, Scientific, and Technical Services 5 Management of Companies and Enterprises 2 Administrative and Support and Waste Management and Remediation Services 2 Educational Services 3 Health Care and Social Assistance 2 Arts, Entertainment, and Recreation 1 Accommodation and Food Service... | https://arxiv.org/abs/2503.23563v4 |
than those in Table 11, suggesting causal associations among different salary groups stay relatively stable across different ages. The SHDs in Table 11 are often 0, specifically for higher age groups, meaning no differences in the estimated causal associations among the salary groups as they move from one age group to ... | https://arxiv.org/abs/2503.23563v4 |
Age group 8 8 6 2 3 4 4 5 0 18 Figure 2: Estimated directed acyclic graphical connections among different variables with different salary levels and age groups with Employment-Quantity. 19 7 Discussion This paper proposed a projection-posterior-based Bayesian method for DAG estimation in a multi- variatetime-seriessett... | https://arxiv.org/abs/2503.23563v4 |
show that the model parameters are continuously iden- tifiable if Dis known as D0. Lemma 2. Let(W0,D0, f0,j:j= 1, . . . , p )and(W,D0, fj:j= 1, . . . , p )be two sets of parameters with the same diagonal matrix, where W0is strictly lower triangular and Whas all diagonal entries zero. If the precision matrices for proce... | https://arxiv.org/abs/2503.23563v4 |
the time series (I−W0)˜Ytare independent. By the assumption, the covariance matrix of (I−W1)˜Yt= (I−W1)(I−W0)−1(I−W0)˜Ytis diagonal. LetA= (I−W1)(I−W0)−1= ( (aj,k) ), which is non-sigular. Then AD 1S1D1AT=D0S0D0 is a diagonal matrix, where S1andS0are the diagonal covariance matrices of the vector of Fourier transforms ... | https://arxiv.org/abs/2503.23563v4 |
of Yj(t)is given by λj(ω) =P k=1dka2 j,kfk(ω),j= 1, . . . , p, that is, (λ1(ω), . . . , λ p(ω))T= (A⊙A)D(f1(ω), . . . , f p(ω))T. Since A⊙Ais assumed to be non-singular and Dhas all entries positive, it follows that the set of functions {λ1(ω), . . . , λ p(ω)}is linearly independent if and only if {f1(ω), . . . , f p(ω... | https://arxiv.org/abs/2503.23563v4 |
Dneed not be known. As the posterior for Ω contracts at Ω0and the map Ω7→(W,D, fj:j= 1, . . . , p )is identifiable by Lemma 3, it remains to show the continuity of the map. For a matrix A, let diag (A)stand for the vector of its diagonal elements. Consider the forward map htaking the value LT((I−W)TD−1(I−W)) = ( ( eT i... | https://arxiv.org/abs/2503.23563v4 |
Applied Economics , 36(1):1–22, 2004. Kevin Bello, Bryon Aragam, and Pradeep Ravikumar. DAGMA: Learning DAGs via m-matrices and a log-determinant acyclicity characterization. Advances in Neural Information Processing Systems, 35:8226–8239, 2022. Emanuel Ben-David, Tianxi Li, Helene Massam, and Bala Rajaratnam. High dim... | https://arxiv.org/abs/2503.23563v4 |
Lauritzen. Graphical Models . Clarendon Press, Oxford, 1996. Kuang-Yao Lee and Lexin Li. Functional structural equation model. Journal of the Royal Statistical Society Series B: Statistical Methodology , 84(2):600–629, 2022. Lizhen Lin and David B Dunson. Bayesian monotone regression using Gaussian process projection. ... | https://arxiv.org/abs/2503.23563v4 |
On Finite Time Span Estimators of Parameters for Ornstein-Uhlenbeck Processes Jun S. Han, Nino Kordzakhia School of Mathematical and Physical Sciences, Macquarie University (email: jun.han1@hdr.mq.edu.au, nino.kordzakhia@mq.edu.au) Abstract We study the bias and the mean-squared error of the maximum likelihood estimato... | https://arxiv.org/abs/2503.23677v1 |
error (MSE) of the maximum likelihood estimators θTofθ= (λ, α), which are defined as bias(θT) =Eλ,α[(θT−θ)], mse (θT) =Eλ,α[(θT−θ)2], with the assumption that the sample size Tis fixed. 2 Studies on bias of estimators in autoregressive processes began from the classical study of the esti- mation of the correlation coef... | https://arxiv.org/abs/2503.23677v1 |
λ∈Λ andα∈ A, where Λ and Aare open intervals in R, the following conditions hold: Pλ,αZT 0|bt|dt <∞ =Pλ,αZT 0σ2 tdt <∞ = 1, (3) Pλ,αZT 0σ−2 tb2 tdt <∞ = 1. (4) While these conditions hold, the measures Pλ,α, λ∈Λ, α∈ Aare equivalent, and this indicates that for any parameters λ∈Λ, λ0∈Λ, α∈ Aandα0∈ A, Pλ,α(A) =Eλ0,... | https://arxiv.org/abs/2503.23677v1 |
αT, we may obtain the modified MLE of λ: ¯λT=ˆλT(¯αT) =¯αTRT 0σ−2 sdYs−RT 0σ−2 sbsdYs QT(¯αT). (26) This MLE ¯λTis connected to the previous MLE ˆλT(α) in (13) via the following relation: ¯λT=ˆλT(α)QT(α) QT(¯αT)+ (¯αT−α)RT 0σ−2 sdYs QT(¯αT). (27) If ¯αTis a strongly consistent estimator, then as T→ ∞ , it can be expect... | https://arxiv.org/abs/2503.23677v1 |
Yis strongly ergodic, and YTd−→Y∞∼ N α,1 2λ . A very well-known explicit solution for the SDE in (29) is given by Yt=α+ (y−α)e−λt+e−λtZt 0eλsdWs, and this indicates that Yis a Gaussian process with the following moments: Eλ,α[Yt] =α+ (y−α)e−λt, V ar λ,α[Yt] =1 2λ(1−e−2λt), Eλ,α[(Yt−α)2] = (y−α)2e−2λt+1 2λ(1−e−2λt), C... | https://arxiv.org/abs/2503.23677v1 |
z1∈R, z2∈R, v≥0, µ≥0, λ > 0. By finding the MGF of the random variable ζ(x) through ψ, one can numerically find Pλ,αn ˆλT(α)< xo through a proper inversion formula, such as the Gaver-Stehfest algorithm in [19]. Other methods in- clude considering the analytical continuation of ψλ,α(z1, z2, v, µ) to the region of comple... | https://arxiv.org/abs/2503.23677v1 |
+µ∂2 ∂λ2ψλ,α(0,0,0, µ)|λ=0 dµ =C1 T2(1 +oT(1)), (50) where C0= 1.7814...andC1= 13.2857.... Similar arguments can be applied to obtain the second order asymptotic expansions for bias(ˆλT(α)) and mse(ˆλT(α)) for the case when λ=u T, T→ ∞ , where uis a constant. 3.4 Properties of the MLE ¯λT=ˆλT(¯αT). Using (13), the mod... | https://arxiv.org/abs/2503.23677v1 |
to higher estimation error if the speed of mean- reversion is faster. At large T, the impact of both λandαon MSE is reduced, suggesting that sample size is the dominant factor in estimation accuracy. In contrast, the non-ergodic case displays qualitatively different behaviour. When λ=−1, we see that both T×bias(ˆλT(α))... | https://arxiv.org/abs/2503.23677v1 |
includes incorporating fractional Ornstein-Uhlenbeck processes that allow for long-range dependence through fractional Brownian motion. Examining the effects of discretisation on the bias and MSE of parameter estimators in these general settings could enhance the understanding of inference under relaxed assumptions, wh... | https://arxiv.org/abs/2503.23677v1 |
and jump detection for drift–diffusion processes. Journal of Econometrics 217, 259–290. [22] Le Breton, A., Pham, D.T., 1989. On the bias of the least squares estimator for the first order autoregressive process. Annals of the Institute of Statistical Mathematics 41, 555–563. [23] Liptser, R.S., Shiryaev, A.N., 2001a. ... | https://arxiv.org/abs/2503.23677v1 |
z2∈R, v≥0, µ≥0. Theorem 3. Letλ >0, Y0=y. Then, ψλ,α(z1, z2, v,µ) =Dλ,α(T, y) expmr+m2q+r2d2/2 1−2qd2 (1−2qd2)−1/2, (55) where Dλ,α(T, y) = exp (λ0α0−λα)y+λ0−λ 2y2−T 2 α2λ2+λ−λ2 0α2 0−λ0 , λ0=p λ2+ 2µ, α 0=λ2 0α+z2 λ2 0, r=z1+λα−λ0α0, q=−λ0−λ+ 2v 2, m=Eλ0,α0[YT] =α0+ (y−α0)e−λt, d2=V ar λ0,α0[YT] =1 2λ0(1−e−λ0T).... | https://arxiv.org/abs/2503.23677v1 |
bivariate-normal distribution, (ξ1, ξ2)∼N=MultinormalDistribution [m1, m2,{{d1, d12},{d12, d2}], we find the MGF using the following command: Qλ,α(u1, u2, µ, u 3) =Integrate [exp{u1x1+u2x2−µ(x2 2−u3x2 1)}PDF[N,{x1, x2}],{x1,−∞,∞},{x2,−∞,∞}]. Now, setting ξ1=YT, ξ2= ¯αT, λ0=p λ2+ 2µ, α 0=λ2α+z2 λ2 0, along with Y0= 0 an... | https://arxiv.org/abs/2503.23677v1 |
Finite sample valid confidence sets of mode Manit Paul1and Arun Kumar Kuchibhotla2 1Department of Statistics & Data Science, University of Pennsylvania 2Department of Statistics & Data Science, Carnegie Mellon University Abstract Estimating the mode of a unimodal distribution is a classical problem in statistics. Altho... | https://arxiv.org/abs/2503.23711v1 |
high probability region, which is found by determining the interval of length an(some sequence) containing the highest number of observations. A modified version of this method is considered in Robertson and Cryer [1974], where for a given function kpnq, the smallest interval contaning kpnqmany observations is selected... | https://arxiv.org/abs/2503.23711v1 |
we do not explore asymptotically valid confidence 2 interval methods like bootstrap or subsampling in this paper. The main contributions of this paper are as follows: 1. Given a data-set of niid observations from a unimodal univariate distribution, we propose novel methods of constructing finite sample valid confidence... | https://arxiv.org/abs/2503.23711v1 |
3 a concave function in the interval rθ0,8q. Then the confidence interval returned by Algorithm 1 satisfies the following: For all ně1and for any αPp0,1q,PFpθ0PxCIn,αqě1´α. Theorem 1 establishes the finite sample validity of the confidence set proposed in Algorithm 1. It is important to note that by construction the co... | https://arxiv.org/abs/2503.23711v1 |
that Width pxCIn,αqshrinks to 0 at the rate of n´1{p1`2βqplognq1{β without any knowledge of the regularity parameter β. The assumption on αp¨,¨,¨qbasically means that near the mode, the density behaves like a power function with exponent β. The main idea of the proof of the theorem is to compare the true weights FpInqw... | https://arxiv.org/abs/2503.23711v1 |
hotla [2025]. The detailed proof of Theorem 3 is provided in Appendix S.4. Remark 2. We have used Hoeffding type bounds to construct the confidence set of θhin Algo- rithm 2. It is possible to obtain better (of smaller width) confidence set by using tighter bounds (such as Bernstein) in the M-estimation problem. We stu... | https://arxiv.org/abs/2503.23711v1 |
The proof of Theorem 5 is very similar to that of Theorem 3. We find a simultaneously valid confidence set of θhfor all hą0 and use that to obtain a valid confidence set of θ0. The detailed proof is provided in Appendix S.6. As in the earlier sections we analyze the width of the confidence set xCSn,αunder assumption 1 ... | https://arxiv.org/abs/2503.23711v1 |
distribution with a unique mode θ0such that EFrlogp1` |pX´θq{pX´θ0q|qsexists for all θPR. Letpθ1be a consistent estimator of θ0. Then we have, xCSEdp n,αa.s.Ñ# θPRˇˇˇˇˇEF„ logˆ 1`ˇˇˇˇX´θ X´θ0ˇˇˇˇ˙ȷ ă1`log 2+ asm´n, nÑ8, The proof of Theorem 8 follows by an application of the law of large numbers. For the complete proof... | https://arxiv.org/abs/2503.23711v1 |
is homogeneous of degree 1. This implies that πSpX´θ0q“πSpU1{γZq“U1{γπSpZq. In other words,rπSpX´θ0qsγ“UrπSpZqsγwhere Uis a uniform-p0,1qrandom variable independent of the random variable rπSpZqsγ. Using the characterization in Result 1, we conclude that rπSpX´θ0qsγ follows unimodal distribution about 0 on R. 10 An imm... | https://arxiv.org/abs/2503.23711v1 |
distribution of the width of the confidence sets at level 1 ´α“0.95. n = 1000 n = 2000 0.940.960.981.00coverage 1.001.251.501.752.001.001.251.501.752.001.002.003.00 βMedian of Width Method M1M2M3Oraclen = 1000 n = 2000 11.21.41.61.8211.21.41.61.821234 βWidth Method M1M2M3Oracle Figure 1: Comparison of coverage and widt... | https://arxiv.org/abs/2503.23711v1 |
Trans- actions on Information Theory , 66(10):6297–6302, 2020. Jiˇ ri Andˇ el. Confidence intervals for the mode based on one observation. Zbornik radova Filozofskog fakulteta u Niˇ su. Serija Matematik , pages 87–96, 1991. Ery Arias-Castro, Wanli Qiao, and Lin Zheng. Estimation of the global mode of a density: Minimax... | https://arxiv.org/abs/2503.23711v1 |
2009. Pascal Massart. The tight constant in the dvoretzky-kiefer-wolfowitz inequality. The annals of Probability , pages 1269–1283, 1990. 14 Abdelkader Mokkadem and Mariane Pelletier. The law of the iterated logarithmfor the multivariate kernel modeestimator. ESAIM: Probability and statistics , 7:1–21, 2003. Richard A ... | https://arxiv.org/abs/2503.23711v1 |
mode estimator for truncated WOD data. In this section we leverage Edelman’s result (2) to construct a finite sample valid confidence set of the mode of a unimodal univarite distribution based on data which may have arbitrary dependency between the data-points. The method is described in Algorithm 6. Algorithm 6: (M3’)... | https://arxiv.org/abs/2503.23711v1 |
unimodal distribution function, for two adjacent disjoint intervals I1“ ra, bs, I2“ rb, cs(where ´8ă aăbăcă8) ifFpI1q{|I1|ąFpI2q{|I2|then either the mode θ0PI1orθ0ăai.e. the mode is ”closer” to the interval I1as compared to I2. Thus we can drop I2and the intervals present after I2from the collection of intervals that c... | https://arxiv.org/abs/2503.23711v1 |
that the following holds. rXKpnq, XKpnq`rpnqs “IBprqifor some i, rXKpnq, XKpnq`rpnqsĂrc, dsand, XKpnq`rpnq´XKpnq“mintWidthpIBprqiq|IBprqiĂrc, dsu. Ifrc, dsdoes not contain rpnq`1 observations then Kpnqcan be set to any arbitrary number (does not affect the proof). It is important to note that by definition rXKpnq, XKpn... | https://arxiv.org/abs/2503.23711v1 |
“p1`opn´β{p1`2βqlogpnqqqp1`opn´1{2qq “1`opn´β{p1`2βqlogpnqq. Suppose Ω Lis the set where our claim that XLpnq“c`opδpnqqwith probability 1 does not hold true. If ωPΩL, there exists a subsequence tnpjquand and ϵsuch that XLpnpjqqąc`R2ϵδpnpjqq for all j. Setting Sn“XLpnq, Tn“XLpnq`rpnqandSn“XJpnq, Tn“XJpnq`rpnqand using L... | https://arxiv.org/abs/2503.23711v1 |
sample mean respectively. Then for αPp0,1qwe have, P˜ pµ´µďp|t1|`|t2|`|t3|qc 3 2n”a logp1{αq`2ı¸ ě1´α. Using Lemma 3 with pt1, t2, t3q“p´ 1{2h,0,1{2hqwe get that tpα, θh,pθ1q“a 3{p2nqp1{hqra logp1{αq` 2ssatisfies the condition (E.7). Hence, xCSh n,α“˜CSh n,α˘h “tθPR|Ln,αpθ,pθ1;S2qď0u˘h “tθPR|Pnpmθ;h´mpθ1;hqďtpα, θ,pθ1q... | https://arxiv.org/abs/2503.23711v1 |
mPMθ0,δ,h,Erm2sďt2ErF2s“t2.1 where t“?C1δ. It can be easily seen that for any εą0, the ε´bracketing covering number Nrspε, M θ0,δ,h,|¨|qď 3. Thus the bracketing entropy integral is, Jrspδ, M θ0,δ,h,|¨|q“żδ 0b 1`logpNrspε, M θ0,δ,h,|¨|qqdεďa 1`log 3δ:“C2δ. Using Theorem-2 .14.17 of Wellner et al. [2013] we have, Ersup m... | https://arxiv.org/abs/2503.23711v1 |
P1˜ DθPRˇˇˇˇˇRMă|θ´θ0|ďh0´h;hc0 4|θ´θ0|βď´n´1{2Gnpmθ;h´mθ0;hq¸ ďP1˜ DθPRˇˇˇˇˇ2M{βphc0{2q´1{βr2{β nă|θ´θ0|ďh0´h;hc0 4|θ´θ0|βď´n´1{2Gnpmθ;h´mθ0;hq¸ ďP1˜ DθPJiforiěMˇˇˇˇˇhc0 4|θ´θ0|βďn´1{2|Gnpmθ;h´mθ0;hq|¸ ď8ÿ i“MP1˜ DθPJiˇˇˇˇˇhc0 4|θ´θ0|βďn´1{2|Gnpmθ;h´mθ0;hq|¸ ď8ÿ i“MP1ˆhc0 42iphc0{2q´1r2 nďn´1{2sup θPJi|Gnpmθ;h´mθ0;hq|... | https://arxiv.org/abs/2503.23711v1 |
if θ0“arg maxxfpxq andθh“arg maxyf˚ghpyq“arg minyPmy;hthen, θ0Ppθh´h, θh`hq. Thus if we can compute a finite sample valid p1´αqconfidence set ˜CIh n,αofθh, then ˜CIh n,α˘hwill be a finite sample valid p1´αqconfidence set of θ0. We use the famous DKW inequality [Massart [1990]] for this purpose, P˜ sup xPR|Pnpp´8 , xsq´... | https://arxiv.org/abs/2503.23711v1 |
1{p1`2βqq `Cp2{c0q1{βC1{βA1{β ϵℓpnq´1´p1{2βq. This completes the proof that n1{p1`2βqℓpnq´1WidthpxCSn,αq“OPp1q. S.8 Proof of Theorem 7 We know from Edelman’s result (2) that for all XiPS2, Pˆ Xi´ˆ2 α´1˙ |Xi´pθ1|ďθ0ďXi`ˆ2 α´1˙ |Xi´pθ1|˙ ě1´α. (E.16) The above result holds because pθ1is independent of any XiPS2. The abov... | https://arxiv.org/abs/2503.23711v1 |
Distributional regression with reject option Clément Dombry∗ Université Marie et Louis Pasteur, CNRS, LmB (UMR 6623), F-25000 Besançon, France. clement.dombry@univ-fcomte.frAhmed Zaoui† Université Marie et Louis Pasteur, CNRS, LmB (UMR 6623), F-25000 Besançon, France. ahmed.zaoui@univ-fcomte.fr Abstract Selective predi... | https://arxiv.org/abs/2503.23782v1 |
estimation of breast cancer ODX scores with uncertainty quantification (Al Masry et al., 2024). Distributional regression models, in particular, generally rely on the minimization of a proper scoring rule to align the predictive distribution with the actual observations. Among these scoring rules, the Continuous Ranked... | https://arxiv.org/abs/2503.23782v1 |
first introduce the framework of distributional regression with reject option, inspired by regression with reject option (Zaoui et al., 2020). The optimal predictor subject to a bounded rejection rate is derived and we show that the reject option must be used if the entropy function associated with the CRPS exceeds som... | https://arxiv.org/abs/2503.23782v1 |
here by its theoretical risk based on the Continuous Ranked Probability Score. 2.1 The Continuous Ranked Probability Score The Continuous Ranked Probability Score (CRPS, Matheson and Winkler 1976) is a widely used scoring rule for assessing the quality of probabilistic forecasts. It quantifies the closeness of a foreca... | https://arxiv.org/abs/2503.23782v1 |
by Γ∗ λ(X) =F∗ XifEnt(F∗ X)≤λ, re otherwise . We can remark that the entropy Ent(F∗ X)plays a central role when deciding whether to use the reject option or not in distributional regression. Whether to make an estimation or not depends on the thresholding of Ent(F∗ X). The following proposition gives the properties of... | https://arxiv.org/abs/2503.23782v1 |
entropy function Ent(F∗ X)around the threshold λε. 2.4 Data-driven procedure The optimal predictor Γ∗ εdepends on the unknown function, the conditional distribution F∗ X, the entropy Ent(F∗ X)and its distribution function. Its estimation naturally relies on a semi-supervised plug-in approach. First, we introduce two in... | https://arxiv.org/abs/2503.23782v1 |
that the divergence term is bounded by the Wasserstein distance of order 1, and the ℓ1-norm of dEntcan similarly be controlled by this distance (see Lemma 3), which offers a graceful measure of discrepancies between distributions. This leads us to the following result. Corollary 1. Letε∈(0,1). Under Assumption 1- 2, it... | https://arxiv.org/abs/2503.23782v1 |
respect to kleads to a minimax rate of convergence of order n−h/(2h+d), see Theorem 3 in Dombry et al. (2024). Recall that for the distributional regression with reject option, the excess risk of our approach is bounded thanks to the Wasserstein distance. Thus, applying Corollary 1 and Theorem 2, we obtain the followin... | https://arxiv.org/abs/2503.23782v1 |
of DRF. Distributional Random Forests (DRF). Random Forest (Breiman, 2001) is a powerful machine learning algorithm in the supervised learning category, suitable for both classification and regression tasks. It works by creating a collection of decision trees, each trained on different random subsets of the data. The f... | https://arxiv.org/abs/2503.23782v1 |
and the rejection rate is fixed, the optimal rule was given and we have shown in particular that the reject option must be used when the entropy function of the CRPS exceeds some threshold. Using the plug-in principle, we have proposed to estimate the optimal rule with an approach applicable to any estimator of the con... | https://arxiv.org/abs/2503.23782v1 |
LmB, UMLP. References Al Masry, Z., Pic, R., Domby, C., and Devalland, C. (2024). A new methodology to predict the oncotype scores based on clinico-pathological data with similar tumor profiles. Breast Cancer Res Treat , 203:587–598. Baran, S. and Lerch, S. (2015). Log-normal distribution based ensemble model output st... | https://arxiv.org/abs/2503.23782v1 |
Statistics. Springer New York. V ovk, V . (2002). On-line confidence machines are well-calibrated. In Proceedings of the Forty-Third Annual Symposium on Foundations of Computer Science , pages 187–196. CA. IEEE Computer Society, Los Alamitos. 11 V ovk, V ., Gammerman, A., and Saunders, C. (1999). Machine-learning appli... | https://arxiv.org/abs/2503.23782v1 |
=nX i=1nX j=1wni(X)wnj(X)(Yj−Yi)1Yi<Yj. Proof. The definition of the weights {wni(X)}ensures that 1−ˆFn,X(Y) =nX j=1wnj(X)1Yj>y. From this, we express Ent( ˆFn,X)as Ent( ˆFn,X) =nX i=1nX j=1wni(X)wnj(X)Z R1Yi≤y1Yj>ydy. The integration over ycan be performed by analyzing the indicator functions 1Yi≤yand1Yj>y. If Yi≤yand... | https://arxiv.org/abs/2503.23782v1 |
λε′) ≥0. Therefore, we can deduce that Err(Γ∗ λε)≤Err(Γ∗ λε′)≤Err(Γ F), and we conclude the result. Proof of Proposition 4. The risk Rλ(ΓF)can be expressed as Rλ(ΓF) =EX Div(FX, F∗ X)1{ΓF(X)̸=re} +EX (Ent( F∗ X)−λε)1{ΓF(X)̸=re} +λε. Then we can deduce that Eλε(ΓF) =EX Div(FX, F∗ X)1{ΓF(X)̸=re} +EX (Ent( F∗ X)−λ... | https://arxiv.org/abs/2503.23782v1 |
• Case 2:{Γ∗ ε(X)\˜Γε(X)}, we have that dEntζ(F∗ X)>˜λεandEnt(F∗ X)≤λε. Thus, Ent(F∗ X)−λε =λε−Ent(F∗ X) =λε−˜λε+˜λε−dEntζ(F∗ X) +dEntζ(F∗ X)−Ent(F∗ X). Using the fact that dEntζ(F∗ X)−˜λε≤0, we get Eh Ent(F∗ X)−λε 1{Γ∗ε(X)\˜Γε(X)}i =Eh (dEntζ(F∗ X)−Ent(F∗ X)) + ( λε−˜λε) 1{Γ∗ε(X)\˜Γε(X)}i . Recall that r(˜Γε) =r(Γ∗ ... | https://arxiv.org/abs/2503.23782v1 |
arXiv:2503.23940v1 [math.PR] 31 Mar 2025OPERATOR LIMIT OF WIGNER MATRICES I DEBAPRATIM BANERJEE Abstract. We consider the Wigner matrix Wnofdimension n×nasn→ ∞. The objective of this paper is two folds: first we construct an operator Won a suitable Hilbert space Hand then define a suitable notion of convergence such that... | https://arxiv.org/abs/2503.23940v1 |
matrix X=/bracketleftbigg 0 1 1 0/bracketrightbigg .This matrix has eigenvalues ±1 when it is defined from R2→R2. However, if we define the operator on the one dimensional subspace/parenleftbigg c 0/parenrightbigg , then/bracketleftbigg 0 1 1 0/bracketrightbigg/parenleftbigg c 0/parenrightbigg =/parenleftbigg 0 c/parenri... | https://arxiv.org/abs/2503.23940v1 |
n. (4) The limit of the Wigner matrices will be denoted by Wand the Hilbert space on which W is defined will be denoted by H. The paper has the following five parts. In the first part, we conside r the Wigner matrices as Kernel and argue that if we want to define the limit of the operator, then th e space must be a non-tri... | https://arxiv.org/abs/2503.23940v1 |
operator (provided it exists) is contained in the space. However in t his section we prove that the spaceL2[0,1] can not be candidate for this space. In particular the action of Knon constant functions gives rise to vectors which in the limiting sense can’t be meas urable functions. We at first give an example regarding... | https://arxiv.org/abs/2503.23940v1 |
continuous. Definition 3is flexible enough to definedF (dx)αfor any 0<α<1 also. It is defined as follows Definition 4. LetFbe a function. Then we define for any 0 <α<1. dF (dx)α((a,b]) =F(b)−F(a) (b−a)α. Hence we can definedB√ dxas well. We shall later identify the limit of Kn,1(f) asdB√ dx. Remark 1. Given any measurable fun... | https://arxiv.org/abs/2503.23940v1 |
2 then this limit exists and is equal to 0. In other words, we should getdF(x)√ dx= 0. This is analogous to the casedF(x) dx= 0 whenever F(x) is a constant function (constant functions have higher order of smoothness than differentiable functions). This phenomenon sh ould reflect in the set function OPERATOR LIMIT OF WIG... | https://arxiv.org/abs/2503.23940v1 |
Suppose we consider two measurable functions fandgsuch that the set functions/integraltextb af(x)dx (b−a),/integraltextb ag(x)dx (b−a)are equivalent. Then Proposition 2proves that they are almost surely same. By Proposition 1, it is enough to consider the case/integraltextb af(x)dx (b−a)=/integraltextb ag(x)dx (b−a). P... | https://arxiv.org/abs/2503.23940v1 |
dB(x)√ dx/parenrightBig2 dx= 1with probability 1. OPERATOR LIMIT OF WIGNER MATRICES I 11 Proof.We at first prove the second part. We take a partition P= (0 =a0< a1< ... < a l= 1) and we write the Riemann type sum for the integral/integraltext1 0/parenleftBig dB(x)√ dx/parenrightBig2 dx. It is given by l/summationdisplay... | https://arxiv.org/abs/2503.23940v1 |
is to incorporate random vectors like fn= (a1,...,a n) such that ai’s are i.i.d. standard Gaussian. We have discussed earlier that the limit of fnmight not be a measurable function. However the ||fn||2= (1+o(1)) almost surely. We next give a definition of convergence offn. Definition 7. For any step function fnwhich is c... | https://arxiv.org/abs/2503.23940v1 |
h is stated below. Theorem 1. LetX1,...,X n,...be a sequence of i.i.d. mean 0variance 1random variables. Define the functions W(n)(t)∈C[0,1]in the following way. If t=i nfor somei∈ {0,...,n}then W(n)(t) =/summationtexti j=1Xj√notherwise linearly interpolate between W(n)(i n)andW(n)(i+1 n)to get the value of 14 DEBAPRATI... | https://arxiv.org/abs/2503.23940v1 |
several different set functions Gcan be equivalentdF√ dx yet/integraltextb aG(x)√ dxmight not be defined. As we are working with bounded operators , we are at liberty to consider only one 0vector. This is the set function where 0((a,b]) = 0for all(a,b]. Hence, we take the unique candidate in each equivalence class anddB(... | https://arxiv.org/abs/2503.23940v1 |
word w=s1···sk, we letGw= (Vw,Ew) be the graph with set of vertices Vw=supp(w) and (undirected) edges Ew={{si,si+1},i= 1,...,k−1}. The graph Gwis connected since the word wdefines a path connecting all the vertices of Gw, which furtherstartsandterminatesatthesamevertex ifthewor disclosed. Wenotethatequivalent words gene... | https://arxiv.org/abs/2503.23940v1 |
random varia blesXn,1,...,X n,m. In particular takingXn,1=...=Xn,m=Yn,1will give the distributional convergence of Yn,1toN(0,Σ(1,1)). Next we state a lemma which is important for proving the asymptotic n ormality. 18 DEBAPRATIM BANERJEE Lemma 4. Suppose we have a sentence aconsisting of mwordsw1,...,w msuch that none o... | https://arxiv.org/abs/2503.23940v1 |
c ovariance matrix is same as covariance of the limiting quantity. Proof of joint asymptotic normality: This proof follows from applications of Theorem 7and OPERATOR LIMIT OF WIGNER MATRICES I 19 Lemma4. We choose finitely many intervals ( c1,d1],...(ck,dk] we call them Q1,...,Q k. We prove (XQi,l1,P1,n,XQi,l2,P2,n)1≤i≤... | https://arxiv.org/abs/2503.23940v1 |
of all observe that in#{(wη(s,1),wη(s,2)) :wη(s,1)=wη(s,2)} nl(Γη(s,1)) , l(Γη(s,1)) =l(Γη(s,2)). Otherwise the count is 0. Let wη(s,1)= (i0,i1,...,i l(Γη(s,1))−1) be a typical word. Observe that the choices for each ijfor 1≤j≤l(Γη(s,1))−2 is (1+o(1))n. Hence the total number of choices for these vertices is (1+ o(1))n... | https://arxiv.org/abs/2503.23940v1 |
So 2( |Ew| −1) + 3≤l⇒ |Ew| ≤l−1 2. So,|Vw| ≤l−1 2+1. However, we fixed the initial vertex i. So, total number of free choices of the vertices are bounded byl−1 2. Hence |V3/parenleftbigi n/parenrightbig | ≤nl−1 2 nl 2C=C√n. Here,Cis a universal constant not depending on n. So,||V3||2=1 n/summationtextn i=1V3/parenleftbi... | https://arxiv.org/abs/2503.23940v1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.