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0φspzqds. Similarly the denominator is given by PpT˚ět|Zq“E„ expˆ ´Xżt 0θsφspZqds´żt 0bsφspZqds˙ |Zȷ “pp1´π0pZqq`e´ItpZqπ0pZqqe´IItpZq. From this we obtain that πtpZq“π0pZqe´ItpZq p1´π0pZqq`e´ItpZqπ0pZq “π0pZq π0pZq`p1´π0pZqqeItpZq “1 1`e´logitpπ0pZqqeItpZq“expitplogitpπ0pZqq´ItpZqq. Example 4.6.2 (ACM in a Cox proport... | https://arxiv.org/abs/2502.07906v1 |
S. Ga¨ ıffas, and S. V. Poulsen. tick: a Python library for statistical learning, with an emphasis on Hawkes processes and time-dependent models. Journal of Machine Learning Research , 18(214):1–5, 2018. I. Bald´ e, Y. A. Yang, and G. Lefebvre. Reader reaction to “Outcome-adaptive lasso: Variable selection for causal i... | https://arxiv.org/abs/2502.07906v1 |
Robins. Double/debiased machine learning for treatment and structural parameters. The Econometrics Journal , 21(1):C1–C68, 01 2018. ISSN 1368-4221. M. Chickering, D. Heckerman, and C. Meek. Large-sample learning of bayesian networks is NP-hard. Journal of Machine Learning Research , 5:1287–1330, 2004. A. M. Christgau a... | https://arxiv.org/abs/2502.07906v1 |
1415–1437, 2017. T. R. Fleming and D. P. Harrington. Counting processes and survival analysis , volume 169. John Wiley & Sons, 2011. P. Forr´ e and J. M. Mooij. A mathematical introduction to causality. Lecture notes, 2023. S. S. Franklin, S. A. Khan, N. D. Wong, M. G. Larson, and D. Levy. Is pulse pressure useful in p... | https://arxiv.org/abs/2502.07906v1 |
1999. H. Ichimura. Semiparametric least squares (SLS) and weighted SLS estimation of single- index models. Journal of Econometrics , 58(1-2):71–120, 1993. C.-R. Jiang and J.-L. Wang. Functional single index models for longitudinal data. Annals of Statistics , 39(1):362–388, 2011. C. Ju, S. Gruber, S. D. Lendle, A. Cham... | https://arxiv.org/abs/2502.07906v1 |
model. The Canadian Journal of Statistics , 31(2):115–128, 2003. T. Manrique. Functional linear regression models: application to high-throughput plant phenotyping functional data . PhD thesis, Universit´ e de Montpellier, 2016. T. Manrique, C. Crambes, and N. Hilgert. Ridge regression for the functional concurrent mod... | https://arxiv.org/abs/2502.07906v1 |
conditional independence via quantile regression based partial copulas. Journal of Machine Learning Research , 22(70):1–47, 2021. J. Pfanzagl and W. Wefelmeyer. Contributions to a general asymptotic statistical theory. Statistics & Risk Modeling , 3(3-4):379–388, 1985. D. Pollard. Convergence of stochastic processes . ... | https://arxiv.org/abs/2502.07906v1 |
and Y. Ben- gio. Toward causal representation learning. Proceedings of the IEEE , 109(5):612–634, 2021. T. Schweder. Composable Markov processes. Journal of Applied Probability , 7(2):400– 410, 1970. D. S ¸ent¨ urk and H.-G. M¨ uller. Functional varying coefficient models for longitudinal data. Journal of the American ... | https://arxiv.org/abs/2502.07906v1 |
Farajtabar, L. Song, X. Yang, and H. Zha. Learning time series associated event sequences with recurrent point process networks. IEEE Transactions on Neural Networks and Learning Systems , 30(10):3124–3136, 2019. H. Xu, M. Farajtabar, and H. Zha. Learning Granger causality for Hawkes processes. In Proceedings of The 33... | https://arxiv.org/abs/2502.07906v1 |
The nature of mathematical models Andrea De Gaetano1,2,3,4,† 1CNR-IASI, Consiglio Nazionale delle Ricerche, Istituto di Analisi dei Sistemi ed Informatica, Rome, Italy; 2CNR-IRIB, Consiglio Nazionale delle Ricerche, Istituto per la Ricerca e l’Innovazione Biomedica, Palermo, Italy; 3Department of Biomatics, Óbuda Unive... | https://arxiv.org/abs/2502.07948v1 |
the equation W=kH (1) Notice that at the present stage we are not dealing with any numbers at all, we are at the first stage of idealization of a relationship between a generic concept of height with a generic concept of weight. Our basic question is then: what, in the above equation, is H? The immediate, naive answer ... | https://arxiv.org/abs/2502.07948v1 |
of course depends on having sampled some individuals from the population of interest and having measured on them both the intended predictor variable(s) and the intended criterion or target variable. This procedure, which is inherently computational, is executed within an n-dimensional Euclidean space, referred to as t... | https://arxiv.org/abs/2502.07948v1 |
vector x). In order to distinguish between a random variable and its realization, we typi- cally indicate the (scalar- or vector- valued) random variable upper case letter ( U, U) and its realization as the appropriate lower case letter ( u,u). In order to keep notation consistent we will strive to have indices and dim... | https://arxiv.org/abs/2502.07948v1 |
not need to indicate the predictors explicitly, since they are part of the experimental design and are fixed. We will thus simply write our predictor function as x=x(θ) (= x(U,θ)) We indicate with a "hat" the estimated value of the parameter, ˆθ, to which corresponds a "best" prediction or forecast ˆx=x(ˆθ). We may als... | https://arxiv.org/abs/2502.07948v1 |
u, different curves corresponding to different values of θ. Since by changing the parameter value the prediction x(u,θ)obviously changes, this clearly leads to thinking about estimation as adapting the predicted curve by minimizing some functional of the "errors" (differences between predicted and observed values ofx),... | https://arxiv.org/abs/2502.07948v1 |
value of the observed phenomenon, necessarily lying on the supposedly true expectation surface, at a position indexed by the supposedly true parameter value θ∗ •xois the observation , the observed point in case space, assumed to be gen- erated by the true linear relationship, but to be affected by some observation erro... | https://arxiv.org/abs/2502.07948v1 |
we have again by Pythagoras theorem that ∥ε∥2=∥ξ∥2+∥e∥2=, where ∥ξ∥2=Pn j=1ξ2 jand∥e∥2=Pn i=1e2 i; • notice however that with respect to the new basis ξj= 0∀j∈ {q+ 1, . . . , n }andei= 0∀i∈ {1, . . . , q }, while ξj∼ N(0, σ2), j∈ {1, . . . , q }and ei∼ N(0, σ2), i∈ {q+ 1, . . . , n } From the above follow important sta... | https://arxiv.org/abs/2502.07948v1 |
are i.i.d. (independent identically distributed), with a normal distribution of zero mean and variance σ2 we have that [2, 3]: 1. there is no guarantee that the local OLS point estimate is the global optimum. 2. approximate asymptotic confidence regions of ˆxonTˆxS, hence of ˆθcan be obtained: such regions would not be... | https://arxiv.org/abs/2502.07948v1 |
H,H=L2(Ω,F, P), this Hilbert space of random variables with finite variance, with ⟨X, Y⟩=E(XY) =R ΩXY dP . Notice that Lp(Ω,F,P)is not a Hilbert space for p̸= 2: we will work strictly in L2(Ω,F, P). 14 Key facts about Hilbert spaces We state in the following a few useful results, proofs can be found in any standard tex... | https://arxiv.org/abs/2502.07948v1 |
easy to see that Mis a vector space with elements in H, hence a subspace of the Hilbert space H=L2(Ω,F,P). As a subspace of a Hilbert space, Min this case inherits its Hilbert space structure (norm, inner product), and a projection operator is guaranteed. Consequently, the point of projection ˆX=PM(Xo)is the one of min... | https://arxiv.org/abs/2502.07948v1 |
the “canonical" chart from the manifold to Rqinduced by the parametrization of the model. As is detailed in Appendix ??We can express the directional derivative of a function fin the direction of a curve γat a point p= γ(λ), with respect to a specific chart map µ, as the dot product of the components of the derivative ... | https://arxiv.org/abs/2502.07948v1 |
We could then hypothesize that the “best” value of θis the value ˆθthat minimizes some kind of distance or divergence from ˜PtoF(θ): in the case the 18 Kullback-Leibler divergence DKLofF(θ)from ˜Pis considered, we obtain the Maximum Likelihood Estimator ˆθML. Now, there is a (biunivocal P-almost-everywhere) relationshi... | https://arxiv.org/abs/2502.07948v1 |
discussion be- cause the underlying manifold is different. As regards information geometry, there is a clear connection since any random variable induces a probability distribution, but the identification of the underlying manifold as determined by the mathemat- ical model of the (biological) experiment, as well as the... | https://arxiv.org/abs/2502.07948v1 |
analysis for probability and stochastic processes,". Cambridge University Press, 2013. [5] P. Klein, "Hilbert spaces and projection theorem", Stock- holm doctoral program in economics. Available at: http://paulklein.ca/newsite/teaching/projections.pdf. [6] Barndorff-Nielsen, O. E., et al. “The Role of Differential Geom... | https://arxiv.org/abs/2502.07948v1 |
Trend estimation for time series with polynomial-tailed noise Michael H. Neumann1Anne Leucht2 1Friedrich-Schiller-Universit¨ at Jena, Institut f¨ ur Mathematik, Ernst-Abbe-Platz 2, D – 07743 Jena, Germany, e-mail: E-mail: michael.neumann@uni-jena.de 2Universit¨ at Bamberg, Institut f¨ ur Statistik, Feldkirchenstraße 21... | https://arxiv.org/abs/2502.08280v1 |
sufficiently strong sense, see e.g. Neumann (1996), Neumann and von Sachs (1997), and Dahlhaus and Neumann (2001). In the context of inde- pendent noise variables which have finite moments of a sufficiently large order, Averkamp and Houdr´ e (2005) showed that soft thresholding achieves the same asymptotic performance ... | https://arxiv.org/abs/2502.08280v1 |
of the linear part and the wavelet coefficients simulta- neously, we use a simpler approach where the linear part is first fitted by least squares and wavelet thresholding is applied to the empirical wavelet coefficients afterwards. Section 5 contains some simulations and a real data example. It is shown that the propo... | https://arxiv.org/abs/2502.08280v1 |
/n)Pn t=1m0(xt)Yt. Under assumption (A2) we obtain by Lemma 7.2 that P |eβj,k−β0 j,k|> t ≤E[(eβj,k−β0 j,k)4] t4≤C(n−1/2/t)4∀t >0,(j, k)∈ In, (2.3) for some C <∞. At fine scales j, 2−3j/2gets smaller than the noise level n−1/2, and the degree of sparsity may be described by the ratio qn,j=n−3j/2/n−1/2. In /Trend estim... | https://arxiv.org/abs/2502.08280v1 |
some γ∈(0,1). Let δ >0, and letπ(N)be the N-fold product of π. LeteT=eT(Y1, . . . , Y N) = (eT1, . . . ,eTn)T be the Bayes estimator of the vector θw.r.t. the prior given by the truncation ofπ(N)to Θ N,q(1+δ). Then a lower bound to the minimax risk over Θ N,q(1+δ)is given by a corresponding Bayes risk, Z ΘN,q(1+δ)Eθh1 ... | https://arxiv.org/abs/2502.08280v1 |
. . , 2j) forms an orthonormal basis of L2((0,1]). An arbitrary function f∈L2((0,1]) can be expanded as f(x) =α0ϕ0(x) +∞X j=02jX k=1βj,kψj,k(x), where α0=R ϕ0(x)f(x)dxandβj,k=R ψj,k(x)f(x)dx. Since the wavelets ψj,1, . . . , ψ j,2jare supported on respective disjoint intervals Ij,1, . . . , I j,2j, we obtain the follow... | https://arxiv.org/abs/2502.08280v1 |
a system of orthogonal functions by eψj,k=1 nj+1,2k−11Ij+1,2k−1−1 nj+1,2k1Ij+1,2k∀(j, k)∈ In, where In:= (j, k):nj+1,2k−1≥1 and nj+1,2k≥1 . Since nJn+1,k< n2−(Jn+1)+ 1≤2, we obtain nJn+1,k≤1 for all k= 1, . . . , 2Jn+1. This implies that (j, k)̸∈ In, ifj > J n, i.e., Jnis the finest scale where functions eψj,kare defi... | https://arxiv.org/abs/2502.08280v1 |
eβj,k)2+ 2tn,j|β|. It follows from (4.3) that E eα0−α0 02 +J∗ n−1X j=0X k: (j,k)∈InE eβj,k−β0 j,k2 =O 2J∗ nn−1 =O n−2/3 (4.7) In order to estimate the contribution to the risk by the other coefficients we use the following result. It improves the simple upper estimateP k: (j,k)∈In(β0 j,k)2∧ t2 n,j≤tn,jP k: (... | https://arxiv.org/abs/2502.08280v1 |
Rosenthal-type inequality (see e.g. Doukhan (1994, page 26, Theorem 2)) that E √n(eβj,k−β0 j,k) γ =E 1√nnX t=1ψj,k(xt)εt γ ≤C E |εt|γ+ϵγ/(γ+ϵ) =:C′′ γ. Then Ggiven by G(x) := 1 −min 1, C′′ γ/xγ satisfies (4.9). Under (A2’) we obtain a rate of convergence without a logarithmic factor which is known to be optima... | https://arxiv.org/abs/2502.08280v1 |
∥bγ−γ0∥2 =O n−1 , 1 nnX t=1 bmn(xt)−m0(xt)2=OP n−2/3log(n) . /Trend estimation with polynomial-tailed noise 16 5. Simulations and data examples 5.1. Simulations We illustrate the finite sample performance of the wavelet estimator with soft thresholding proposed in Section 4.2 using the following two trend functi... | https://arxiv.org/abs/2502.08280v1 |
to the COVID pandemic. We applied the partially linear model with soft thresholding, introduced in Section 4.3 with three different choices for the thresholding parameter K= 0.05, K = 0.1, K = 0.2. For comparison, we additionally fitted a partially linear model, where the nonlinear part is estimated via classical kerne... | https://arxiv.org/abs/2502.08280v1 |
two cases. 1)|θk|< t/2 If|εk|=|Yk−θk| ≤t/2, then |Yk|< tand so bθk= 0. Otherwise, we /Trend estimation with polynomial-tailed noise 20 use the estimate |bθk−θk| ≤ |bθk−Yk|+|Yk−θk| ≤t+|εk| ≤3|εk|. Therefore, E (bθk−θk)2 ≤θ2 k+ 9E ε2 k 1(|εk|> t/2) . 2)|θk| ≥t/2 In this case we use that |bθk−θk| ≤t+|εk|, which implie... | https://arxiv.org/abs/2502.08280v1 |
extreme case in order to provide some intuition. If m0= 1[c,xn]for any cbetween x1and xn, then we obtain that at each scale jonly one of the coefficients β0 j,kcan be non-zero with β0 j,k=O(2−j/2). Since 2−j/2≥tn,jif and only if 2j≤n2/3, we obtain that Sn m0 ≤X j:n1/3≤2j≤n2/3t2 n,j+X j: 2j>n2/3X k: (j,k)∈In(β0 j,k)2 ... | https://arxiv.org/abs/2502.08280v1 |
Dahlhaus, R. and Neumann, M.H. (2001). Locally adaptive fitting of semi- parametric models to nonstationary time series. Stoch. Proc. Appl. 91(2), 277–308. Donoho, D.L. (1995). De-noising by soft-thresholding. IEEE Transactions on information theory 41(3), 613–627. Donoho, D.L. and Johnstone, I.M. (1994). Ideal spatial... | https://arxiv.org/abs/2502.08280v1 |
A comparison of Dirichlet kernel regression methods on the simplex Hanen Daayeba, Christian Genestb, Salah Khardania, Nicolas Klutchnikoffc, Fr´ ed´ eric Ouimetb aD´ epartement de math´ ematiques, Universit´ e de Tunis El-Manar, 2092 El Manar 2, Tunis, Tunisia bDepartment of Mathematics and Statistics, McGill Universit... | https://arxiv.org/abs/2502.08461v1 |
the fixed kernel by replacing it with a so-called boundary kernel that solved a specific variational problem near the boundary. The estimator was shown to perform better than the estimator proposed by Priestley and Chao (1972) and the classical Nadaraya–Watson estimator (Nadaraja, 1964; Watson, 1964), proposed a few ye... | https://arxiv.org/abs/2502.08461v1 |
setting, one can consider products of univariate kernels on product spaces (Bouezmarni and Rombouts, 2010; Kokonendji and Som´ e, 2018), or kernels that are tailored to non-product spaces such as Dirichlet kernels on the simplex (Ouimet and Tolosana-Delgado, 2022; Bertin et al., 2023), Wishart kernels on the cone of po... | https://arxiv.org/abs/2502.08461v1 |
∥s∥1. For a given a sample size n∈N, letY1, . . . , Y nbe the response variables associated with a set of known and fixed design points x1, . . . ,xnon the simplex. The design density fcorresponds to the density of the design points in the limit n→ ∞ . Formally, it is defined such that, for any Borel setB∈ B(Rd), lim n... | https://arxiv.org/abs/2502.08461v1 |
min x∈Sdf(x)≥f0and max x∈Sdσ2(x)≤σ2 0. (A3) b=b(n)→0 and b−1n−1/d→0 asn→ ∞ . (A4) There exists a sequence B1, . . . , B nof convex compact sets such that (a) Int( Bi)∩Int(Bj) =∅for all i̸=j, and B1∪ ··· ∪ Bn=Sd; (b)xi∈Int(Bi) for every i∈[n]; (c) the boundary of Bihas Lebesgue measure zero, i.e., λ(∂Bi) = 0; (d) max x,... | https://arxiv.org/abs/2502.08461v1 |
(A4) (d) would need to be relaxed by a logarithmic factor, as shown in Theorem 5.1 of Devroye et al. (2017); see also Gibbs and Chen (2020) for a closely related analysis. In turn, this additional factor in Assumption (A4) (d) would have an impact on the asymptotics of every result in Section 4. Figure 3.1: The black d... | https://arxiv.org/abs/2502.08461v1 |
Proposition 4.1 (Pointwise bias) .Suppose that Assumptions (A1)–(A4) hold. Then, as n→ ∞ and uniformly for all s∈ Sd, one has Bias{ˆm(GM) n,b(s)}=E{ˆm(GM) n,b(s)} −m(s) =b g(s) + o( b) +O(n−1/d), where the function gis defined, for all s∈ Sd, by g(s) =dX i=1{1−(d+ 1)si}∂ ∂sim(s) +1 2dX i,j=1si(1{i=j}−sj)∂2 ∂si∂sjm(s). ... | https://arxiv.org/abs/2502.08461v1 |
kth-order kernels, refer to M¨ uller and Prewitt (1993). Theorem 4.5 (Asymptotic normality) .Suppose that Assumptions (A1)–(A4) hold, and consider a fixed point s∈Int(Sd)such that σ2(s)∈(0,∞). Assume further that the rate Rn=n1/2bd/4 grows to infinity as n→ ∞ andb→0, and that the errors ε1, . . . , ε nin the model (2.1... | https://arxiv.org/abs/2502.08461v1 |
satisfy ε1, . . . , ε niid∼ N 0,1 10IQR{m(x1), . . . , m (xn)} , and IQR( ·) denotes the interquartile range. Bandwidth selection is performed using least-squares cross-validation (LSCV). More specifically, for a given method in {GM,LL,NW}and a target regression function m, the bandwidth ˆbn∈(0,∞) is chosen to minimi... | https://arxiv.org/abs/2502.08461v1 |
al. (2025), is notable for its broad coverage, extensive number of samples, and rigorous analytical techniques. Among the suite of data collected are properties such as elemental concentrations, particle size distribution, and pH levels measured in a calcium chloride (CaCl2) solution. The use of CaCl2 provides a stabil... | https://arxiv.org/abs/2502.08461v1 |
points of the form xi= (xi,1, xi,2), which represent the (renormalized) proportions of sand and silt in each soil sample, respectively. The proportion of clay is determined by the complement, 1 −xi,1−xi,2. The pH in CaCl2 of each sample, denoted yi, serves as an explanatory variable. The goal is to use the LL smoother ... | https://arxiv.org/abs/2502.08461v1 |
in a blue-green shade, suggesting a pH level around 4.5 to 5.0, indicating that the soil is likely very strongly acidic . 7. Moderate sand and silt (e.g., the sum of sand and silt is 60% ±5%): This region is in a light orange shade, suggesting a pH level around 7.5 to 8.0, which would classify the soil as slightly to m... | https://arxiv.org/abs/2502.08461v1 |
Assumption (A1), along with the uniform upper and lower bounds on σ2andffrom Assumption (A2), one sees that, for all t∈Bi, σ2(t) f(t)=σ2(ti) f(ti)+O(n−1/d). Therefore, proving (7.2) reduces to showing that n−1nX i=1σ2(ti) f(ti)λ(Bi)κ2 s,b(ti) =n−1nX i=1σ2(ti) f(ti)Z Biκ2 s,b(t)dt× {1 + o(1) }, which is analogous to (7.... | https://arxiv.org/abs/2502.08461v1 |
Biκs,b(x)dx, where the random variables Zs,b,1, . . . , Z s,b,nare centered and uncorrelated. The asymptotic nor- mality of n1/2bd/4Pn i=1Zs,b,iwill be established by verifying the following Lindeberg condition for double arrays: Whatever δ∈(0,∞), one has, as n→ ∞ , Pn i=1E |Zs,b,i|21{|Zs,b,i|>2δsn,b} s2 n,b→0, (7.7)... | https://arxiv.org/abs/2502.08461v1 |
ebec (www.calculquebec.ca) and the Digital Research Alliance of Canada (www.alliancecan.ca). Credit is given to Raimon Tolosana-Delgado (Helmholtz-Zentrum Dresden-Rossendorf) for suggesting the real-data application in Section 6 and providing the GEMAS dataset. Funding Funding in support of this work was provided by th... | https://arxiv.org/abs/2502.08461v1 |
curve estimation (Proc. Workshop, Heidelberg, 1979) , volume 757 of Lecture Notes in Math. , pages 23–68. Springer, Berlin, 1979. MR564251. 20 T. Gasser, H.-G. M¨ uller, and V. Mammitzsch. Kernels for nonparametric curve estimation. J. Roy. Statist. Soc. Ser. B , 47(2):238–252, 1985. MR564251. C. Genest and F. Ouimet. ... | https://arxiv.org/abs/2502.08461v1 |
estimates. Period. Math. Hungar. , 17(2):83–108, 1986. MR858109. C. J. Stone. Consistent nonparametric regression. Ann. Statist. , 5(4):595–645, 1977. MR443204. C. J. Stone. Optimal rates of convergence for nonparametric estimators. Ann. Statist. , 8(6):1348– 1360, 1980. MR594650. C. J. Stone. Optimal global rates of c... | https://arxiv.org/abs/2502.08461v1 |
arXiv:2502.08510v1 [math.ST] 12 Feb 2025Moment Estimator-Based Extreme Quantile Estimation with Erroneous Observations: Application to Elliptical Extreme Quantile Region Estimation Jaakko Pere∗1, Pauliina Ilmonen1, and Lauri Viitasaari2 1Aalto University School of Science, Finland 2Aalto University School of Business, ... | https://arxiv.org/abs/2502.08510v1 |
Theorem 1, provides error bounds related to extreme quantile esti- mation under approximated observations. Our result highli ghts that extreme quantile estimation is very sensitive to approximation errors, part icularly in the case γ≤0, when decaying approximation error does not automatically guara ntee decay of the es... | https://arxiv.org/abs/2502.08510v1 |
−p)-quantile is related to the smaller (1 −k/n)- quantile by U/parenleftigg1 p/parenrightigg ≈U/parenleftbiggn k/parenrightbigg +a/parenleftbiggn k/parenrightbigg/parenleftig k np/parenrightigγ−1 γ. The role of the integer kis to control the number of tail observations used in the esti - mation, and for asymptotic ... | https://arxiv.org/abs/2502.08510v1 |
applications. Notethatthisfurther implies U(∞)>0whichisneeded fortheestimators to be well-defined. This assumption is usually assumed impli citly in the context of extreme value theory. Theorem 1. LetYbe an almost surely positive random variable with F∈ D(Gγ), γ∈R. LetY= (Y1,...,Y n)be i.i.d. copies of Yand letˆY=/paren... | https://arxiv.org/abs/2502.08510v1 |
hence we take the symmetric square root Σ1/2. Moreover, without the assumption det(Σ) = 1, the generating variate Rand the scatter matrix Σ are unique only up to a positive constant. That is, the parameters ( µ,R,Σ) and/parenleftig µ,1√cR,cΣ/parenrightig ,c >0, define the same model. To guarantee identifiability, typic... | https://arxiv.org/abs/2502.08510v1 |
i.i.d. sample from X. Denote qγ(t) =/integraltextt 1sγ−1lnsdsanddn=k/(np). Assume that k=knsatisfies(A1), and assume that, asn→ ∞, the following conditions hold: (A6)√ kQ(n/k)→λ∈R, whereQis the auxiliary function from the second-order condition for lnUR. (A7)np=npn=O(1)andqγ(dn)//parenleftig dγ n√ k/parenrightig →0. (... | https://arxiv.org/abs/2502.08510v1 |
i=/bardblZi−ˆµ(Z)/bardblˆΣ(Z), ˆR′=/parenleftigˆR′ i,...,ˆR′ n/parenrightig and ˆQ′ p=/braceleftig z∈Rd:/bardblz−ˆµ(Z)/bardblˆΣ(Z)≥ˆxp/parenleftigˆR′/parenrightig/bracerightig . Acknowledgements: Jaakko Pere gratefully acknowledges support from the Vilho , Yrjö and Kalle Väisälä Foundation. Pauliina Ilmonen grate... | https://arxiv.org/abs/2502.08510v1 |
>0. Lemma 3 ([3], Lemma 1.2.9) .SupposeU∈ERVγ. 1. Ifγ >0, thenU(∞) =∞andlimt→∞U(t)/a(t) = 1/γ. 12 2. Ifγ= 0, thenUis slowly varying and limt→∞a(t)/U(t) = 0. 3. Ifγ <0, thenU(∞)<0andlimt→∞a(t) = 0. Lemma 4 ([15], Lemma 3) .Suppose U∈2ERVγ,ρwithγ∈Randρ <0. Then limt→∞t−γa(t) =c∈(0,∞). We next prove our main result, Theor... | https://arxiv.org/abs/2502.08510v1 |
that ( 7) and(A3)imply√ k(ˆγ−˜γ) =OP(1). Moreover, by (A2)we get |(ˆγ−˜γ)lns| ≤/vextendsingle/vextendsingle/vextendsingle√ k(ˆγ−˜γ)/vextendsingle/vextendsingle/vextendsinglelndn√ k=oP(1) for any 1 ≤s≤dn. These imply sup 1≤s≤dn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglee(ˆγ−˜γ)lns−1 (ˆγ−˜γ)lns... | https://arxiv.org/abs/2502.08510v1 |
1√n/parenrightig such that/vextendsingle/vextendsingle/vextendsingleˆR2 n−j,n−R2 n−j,n/vextendsingle/vextendsingle/vextendsingle≤KnR2 n−j,n. This gives Kn≥/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆR2 n−j,n−R2 n−j,n R2 n−j,n/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsi... | https://arxiv.org/abs/2502.08510v1 |
<0, by using U(∞)∈(0,∞) and (26), we obtain√ krp√na(n k)=O/parenleftbigg/radicalig k n1 a(n/k)/parenrightbigg =O/parenleftig√ kzn/parenrightig =o(1). For γ >0, Lemma 3givesU(t)/a(t) =O(1), and Lemma 4and(A8)now provide √ krp√na/parenleftig n k/parenrightig=/radicaligg k nU(1/p) a(1/p)a(1/p) (1/p)γ(n/k)γ a(n/k)/pa... | https://arxiv.org/abs/2502.08510v1 |
Abstract questionnaires and FS-decision digraphs Jiaye Chen, Suzan Kadri, Mateja ˇSajna∗, and Ioana S ,chiopu-Kratina University of Ottawa February 13, 2025 Abstract A questionnaire is a sequence of multiple choice questions aiming to collect data on a population. We define an abstract questionnaire as an ordered pair ... | https://arxiv.org/abs/2502.08522v1 |
tree, however, each question corresponds to a level, and thus may be represented by many vertices, each corresponding to a different sequence of responses to the previous questions. The edges from a given vertex to its children in the tree represent the possible responses to the question. The ordering of the children a... | https://arxiv.org/abs/2502.08522v1 |
0 ,1, . . . , N −1) such that question ihasmipossible answers. We denote the set of possible answers to question ibyAi; usually, we assume that Ai=Zmi. An abstract questionnaire ( N,M) is called binary if M= (2,2, . . . , 2); that is, if every question has exactly two answers. 2.1 Graphs, trees, and digraphs Agraph Gis... | https://arxiv.org/abs/2502.08522v1 |
contains vand all of its descendants. Adirected graph (shortly digraph )Dis an ordered triple ( V, A, ψ ), where Vis a non- empty finite set, Ais a finite set disjoint from V, and ψis a function assigning to each element of Aan ordered pair of elements of V. Sets VandAare the vertex set andarc set , respectively, of th... | https://arxiv.org/abs/2502.08522v1 |
be fixed; that is, independent from the responder and independent from the responses to any previous questions. We will, however, assume that these orderings respect a given precedence relation. In other words, our input will be a binary relation Ron the set of questions ZNsuch that ( a, b)∈Rif question amust be asked ... | https://arxiv.org/abs/2502.08522v1 |
Sthat extend R. ifS=∅thenT=T+ [L] # total order Lis complete; add it to list T else M:= the set of minimal elements of the partial order ( S, R) form∈Mdo L′:=L+ [m] S′:=S− {m} R′:=R∩(S′×S′) TotalOrder( S′, R′, L′,T) 6 4 Representing the flow of a questionnaire by a graph In this section, we aim to represent the flow of... | https://arxiv.org/abs/2502.08522v1 |
we are going to skip all unnecessary vertices; these are vertices corresponding to answer strings that we do not want to be included in the questionnaire. Second, we are going to flag all those vertices whose answer strings are of special interest (possibly contradictory, but should not be excluded). The latter feature... | https://arxiv.org/abs/2502.08522v1 |
. . m i. We define the edge set of T, together with the •subset Uofskipped vertices , •question assignment κ:V(T)→ZN+1and •answer string assignment α:V(T)→ {ϵ} ∪SN−1 i=0A∗ 0×A∗ 1×. . .×A∗ i recursively as follows. (i) Vertex 0 is the root, κ(0) = 0 and α(0) = ϵ. (ii) If uis a vertex with κ(u) =q≤N, then u∈Uif and only ... | https://arxiv.org/abs/2502.08522v1 |
(0 , k)-answer string a=a0. . . a ksuch that a∈S′ k+1−Sk+1. Since a∈S′ k+1, by Lemma 4.6, there is a vertex u∈V(T) such that α(u) =a. However, by the first observation of this proof, we know u∈U, and hence a∈Sk+1, a contradiction. We conclude that S′ k+1=Sk+1. From Definition 4.5, it is also clear that the flag functio... | https://arxiv.org/abs/2502.08522v1 |
. . f ifor some i < ℓ, and by Definition 4.8, we know that the (0 , N−1)-answer string f0. . . f j∗. . .∗ is not in Ffor all j < ℓ . Hence by Definition 4.3, vertex uis not flagged. Letube a vertex of Tthat is either vitself or a descendant of v. Then α(u) = f0. . . f ℓfℓ+1. . . f kfor some k,ℓ≤k≤N−1, and fℓ+1. . . f k... | https://arxiv.org/abs/2502.08522v1 |
of the questionnaire. 4.2.1 Vertex equivalence and FS-decision digraphs The following definition will make the idea of similar subtrees precise. Definition 4.11 LetQ= (N,M) be an abstract questionnaire with flag-set Fand skip-list S, and let Tbe the FS-decision tree for Q, with question assignment κand answer string as... | https://arxiv.org/abs/2502.08522v1 |
a level function κ:V→Nassigning to each vertex vits level κ(v) so that (i) there exists at least one vertex vsuch that κ(v) = 0, and (ii) if ( v, u)∈A, then κ(u) =κ(v) + 1. Note that an ordered rooted tree can be viewed as an ordered levelled digraph with κ being the distance from the root, all arcs being directed away... | https://arxiv.org/abs/2502.08522v1 |
edge away from the root. Any other FS-decision digraph is obtained from DTvia a sequence of operations whereby the subdigraphs rooted at two equivalent vertices are merged while preserving all information encoded by the FS-decision tree. Recall that the list of out-neighbours of a vertex in an FS-decision digraph may c... | https://arxiv.org/abs/2502.08522v1 |
In Theorem 4.18 below, which will be crucial for constructing FS-decision digraphs, we show that two vertices in an FS-decision tree are equivalent if and only if their local flag-sets, 17 FS-decision treeFully reduced FS-decision digraphFS-decision treeFully reduced FS-decision digraphFigure 2: FS-decision trees for b... | https://arxiv.org/abs/2502.08522v1 |
j≤ℓ, the (0 , N−1)-answer string b0. . . b kak+1. . . a j∗. . .∗ ∈F. Note that by the conclusion of the previous paragraph, we indeed have j≥k+ 1. By Lemma 4.9, there exists a vertex z′∈V(T) such that α(z′) =b0. . . b kak+1. . . a j, and necessarily z′∈V(Tw). Let u′= Φ−1(z′). Then α(u′) = a0. . . a kak+1. . . a j, and ... | https://arxiv.org/abs/2502.08522v1 |
that ai+1=∗if and only if u′∈U, that is, if and only if α(u′)∈Si+1, that is, if and only if ak+1. . . a i∈Sa i+1. However, since there is no vertex z′′with α(z′′) =bak+1. . . a iai+1, we have that ai+1=∗if and only ifz′̸∈U, that is, if and only if α(z′)̸∈Si+1, that is, if and only if ak+1. . . a i̸∈Sb i+1. Since Sa i+1... | https://arxiv.org/abs/2502.08522v1 |
reduced FS-decision digraph of an abstract question- naire procedure FS-digraph( N,M, F,S) #Input: abstract questionnaire (N,M)with flag-set F, compatible with the skip-list S. #Output: a fully reduced FS-decision digraph Dfor(N,M)with the subset of skipped vertices U, question assignment κ, answer string assignment A,... | https://arxiv.org/abs/2502.08522v1 |
where ai∈A⋄ ifor all i=k, k+ 1, . . . , ℓ . A generalized ( k, ℓ)-answer string akak+1. . . a i−1⋄ai+1. . . a ℓcan be thought of as repre- senting all ( k, ℓ)-answer strings of the form akak+1. . . a i−1xai+1. . . a ℓ, forx∈Ai. We next explain what we mean by a pre-skip-list. Definition 4.21 Apre-skip-list for the abst... | https://arxiv.org/abs/2502.08522v1 |
respec- tively, and the answers 0 and 1 to questions 3 and 4, respectively, are of special interest. Then our pre-flag-set is P={1⋄2⋄⋄,⋄⋄⋄01}, and it represents the set of answer strings {1a12a3a4:ai∈Ai, i= 1,3,4} ∪ {a0a1a201 :ai∈Ai, i= 0,1,2}. Given a skip-list Sand pre-flag-set P, Algorithm 4.24 below creates a flag-... | https://arxiv.org/abs/2502.08522v1 |
with no directed cycles. By Lemma 3.1, there exists a total order extending R, and we may apply Algorithm 3.2. The resulting output is T= [ [0 ,1,2,3,4],[0,1,2,4,3],[0,1,4,2,3],[0,2,1,3,4],[0,2,1,4,3],[0,2,3,1,4] ]; that is, Tis a list of all total orders on Z5that extend the relation R. Note that for the remainder of ... | https://arxiv.org/abs/2502.08522v1 |
hope is that our models will help questionnaire designers visualize and automatize their work. Acknowledgement The third author gratefully acknowledges support by the Natural Sciences and Engineer- ing Research Council of Canada (NSERC), Discovery Grant RGPIN-2022-02994. References [1] J. Bethlehem and A. Hundepool, TA... | https://arxiv.org/abs/2502.08522v1 |
arXiv:2502.08539v2 [stat.ME] 30 Apr 2025Anytime-valid FDR control with the stopped e-BH procedure Hongjian Wang∗1, Sanjit Dandapanthula1, and Aaditya Ramdas2 1, 2Department of Statistics and Data Science, Carnegie Mellon Univers ity 2Machine Learning Department, Carnegie Mellon University {hjnwang,sanjitd,aramdas }@cmu... | https://arxiv.org/abs/2502.08539v2 |
al. (2021), however, bears a very crucial caveat. It is assumed that the e-processes of the hypotheses are valid under a shared “global” filtration. This indicates that one can notsimply apply e-BH to e-processes each constructed “locally” withinthe hypotheses without further verification. As we shall ela borate soon, fil... | https://arxiv.org/abs/2502.08539v2 |
That is, a random set of rejected hypot heses controls the FDR if and only if it is the e-BH of Gcompound e-values. We shall revisit this equivalence after fully developing our theory in the sequential setting. 2.2 Sequential multiple testing and filtrations We now introduce the sequential experiment setting and let u s... | https://arxiv.org/abs/2502.08539v2 |
{Fn}and sup P∈M τ∈T /summationdisplay g∈[G] /BD{P∈Pg}·EPgMg τ /lessorequalslantG. (9) These definitions satisfy the clear inclusion relations 1⊆2and3⊆4.1and3, for example, are not included in either direction, because while adaptiv ity to a larger filtration is a weaker condition, returningane-value at any stoppingti... | https://arxiv.org/abs/2502.08539v2 |
es in the literature and still enjoy the global property to stop them at a single cross-hypothesi s stopping time. 3 Global e-processes by a Markovian assumption We formulate the sequential multiple testing problem with t he following set-up. Consider at timen= 1,2,..., we observe the nthobservation Zn= (Xn,Yn) that in... | https://arxiv.org/abs/2502.08539v2 |
there is aGn−1-measurable random function Eg n: Θg 0×X×Yg→R/greaterorequalslant0that satisfies sup θ∈Θg 0 x∈X/integraldisplay YgEg n(θ,x,y)pg(dy|x,θ)/lessorequalslant1. (15) Then, under Assumption 3.1, under any θ∗∈Θg 0, the process Mg n(θ∗) =n/productdisplay i=1Eg i(θ∗,Xi,Yg i) (16) is a nonnegative supermartingale on ... | https://arxiv.org/abs/2502.08539v2 |
and Ramdas (2024, Theorem 2) and yields an e-process for any finer filtrations. That is, if{Mn}is an e-process for Pon{Gn}, then for any adjuster Aand any filtration {Fn}⊇{G n}, the adjusted process {Ma n}where Ma 0= 1, Ma n=A/parenleftbigg max 0/lessorequalslanti/lessorequalslantnMi/parenrightbigg (24) 7 is an e-process ... | https://arxiv.org/abs/2502.08539v2 |
hypothesis dependenceamong Y|X. Then, thestepwise e-value function Eg ntakes the likelihood ratio Eg n(θ,x,y) =qg(y|x) pg(y|x), (31) Once again, ( 15) holds with equality and Eg n(θ,x,y) does not dependent on the parameter θ. The corresponding e-process therefore equals the global te st martingale Mg n(θ) with any θ∈Θg... | https://arxiv.org/abs/2502.08539v2 |
Eg ndoes not dependent on the “parameter” θthat now lives in an infinite-dimensional space. The corresp onding e-process therefore equals the supermartingale Mg n(θ) with any θ∈Θg 0: Ug n=Mg n(θ) = exp/braceleftBiggn/summationdisplay i=1φ(λi(Yg i−µg(Xi)))−/summationtextn i=1λ2 ivg(Xi) 2/bracerightBigg . (41) If the null... | https://arxiv.org/abs/2502.08539v2 |
eBHα(Mg τ:g∈[G]), (48) one is essentially applying the BH procedure to Gp-values, BHα(1/Mg τ:g∈[G]), (49) therefore potentially facing the same possible FDR inflatio n that can at worst be 1+2−1+···+ G−1≈logG, astheonearisinginapplyingBHtoarbitrarilydependentp -values(Wang and Ramdas , 2022, Theorem 1). The complication... | https://arxiv.org/abs/2502.08539v2 |
Research Lab, Stanford University, Winter 2023 edition, 2023. S. R. Howard, A. Ramdas, J. McAuliffe, and J. Sekhon. Time-unif orm, nonparametric, nonasymptotic confidence sequences. The Annals of Statistics , 49(2):pp. 1055–1080, 2021. ISSN 00905364, 21688966. URL https://www.jstor.org/stable/27169410 . N. Ignatiadis, R.... | https://arxiv.org/abs/2502.08539v2 |
Network Goodness-of-Fit for the block-model family Jiashun Jin∗ Department of Statistics, Carnegie Mellon University and Zheng Tracy Ke Department of Statistics, Harvard University and Jiajun Tang Department of Statistics, Harvard University and Jingming Wang Department of Statistics, University of Virginia February 13... | https://arxiv.org/abs/2502.08609v1 |
. . . . . . . . . . . 16 3.2 Asymptotic normality of the GoF-MSCORE . . . . . . . . . . . . . . . . . . . 21 3.3 Extension of GoF-MSCORE from DCMM to MMSBM . . . . . . . . . . . . . 24 3.4 GoF-SCORE for DCBM and SBM . . . . . . . . . . . . . . . . . . . . . . . . 25 3.5 Power analysis and optimality . . . . . . . . . .... | https://arxiv.org/abs/2502.08609v1 |
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