Local Dvoretzky-Kiefer-Wolfowitz confidence bands
Type de document :
Compte-rendu et recension critique d'ouvrage
Titre :
Local Dvoretzky-Kiefer-Wolfowitz confidence bands
Auteur(s) :
Maillard, Odalric Ambrym [Auteur]
Inria Lille - Nord Europe
Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 [CRIStAL]
Inria Lille - Nord Europe
Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 [CRIStAL]
Titre de la revue :
Mathematical Methods of Statistics
Éditeur :
Springer
Date de publication :
2022
ISSN :
1066-5307
Mot(s)-clé(s) en anglais :
Cumulative Distribution Function
Concentration inequalities
DKW
Risk measure
Concentration inequalities
DKW
Risk measure
Discipline(s) HAL :
Statistiques [stat]/Théorie [stat.TH]
Statistiques [stat]/Machine Learning [stat.ML]
Statistiques [stat]/Machine Learning [stat.ML]
Résumé en anglais : [en]
In this paper, we revisit the concentration inequalities for the supremum of the cumulative distribution function (CDF) of a real-valued continuous distribution as established by Dvoretzky, Kiefer, Wolfowitz and revisited ...
Lire la suite >In this paper, we revisit the concentration inequalities for the supremum of the cumulative distribution function (CDF) of a real-valued continuous distribution as established by Dvoretzky, Kiefer, Wolfowitz and revisited later by Massart in in two seminal papers. We focus on the concentration of the local supremum over a sub-interval, rather than on the full domain. That is, denoting U the CDF of the uniform distribution over [0, 1] and U n its empirical version built from n samples, we study P sup u∈[u,u] U n (u)−U (u) > ε for different values of u, u ∈ [0, 1]. Such local controls naturally appear for instance when studying estimation error of spectral risk-measures (such as the conditional value at risk), where [u, u] is typically [0, α] or [1 − α, 1] for a risk level α, after reshaping the CDF F of the considered distribution into U by the general inverse transform F −1. Extending a proof technique from Smirnov, we provide exact expressions of the local quantities P sup u∈[u,u] U n (u) − U (u) > ε and P sup u∈[u,u] U (u) − U n (u) > ε for each n, ε, u, u. Interestingly these quantities, seen as a function of ε, can be easily inverted numerically into functions of the probability level δ. Although not explicit, they can be computed and tabulated. We plot such expressions and compare them to the classical bound log(1/δ) 2n provided by Massart inequality. We then provide an application of such result to the control of generic functional of the CDF, motivated by the case of the conditional value at risk. Last, we extend the local concentration results holding individually for each n to time-uniform concentration inequalities holding simultaneously for all n, revisiting a reflection inequality by James, which is of independent interest for the study of sequential decision making strategies.Lire moins >
Lire la suite >In this paper, we revisit the concentration inequalities for the supremum of the cumulative distribution function (CDF) of a real-valued continuous distribution as established by Dvoretzky, Kiefer, Wolfowitz and revisited later by Massart in in two seminal papers. We focus on the concentration of the local supremum over a sub-interval, rather than on the full domain. That is, denoting U the CDF of the uniform distribution over [0, 1] and U n its empirical version built from n samples, we study P sup u∈[u,u] U n (u)−U (u) > ε for different values of u, u ∈ [0, 1]. Such local controls naturally appear for instance when studying estimation error of spectral risk-measures (such as the conditional value at risk), where [u, u] is typically [0, α] or [1 − α, 1] for a risk level α, after reshaping the CDF F of the considered distribution into U by the general inverse transform F −1. Extending a proof technique from Smirnov, we provide exact expressions of the local quantities P sup u∈[u,u] U n (u) − U (u) > ε and P sup u∈[u,u] U (u) − U n (u) > ε for each n, ε, u, u. Interestingly these quantities, seen as a function of ε, can be easily inverted numerically into functions of the probability level δ. Although not explicit, they can be computed and tabulated. We plot such expressions and compare them to the classical bound log(1/δ) 2n provided by Massart inequality. We then provide an application of such result to the control of generic functional of the CDF, motivated by the case of the conditional value at risk. Last, we extend the local concentration results holding individually for each n to time-uniform concentration inequalities holding simultaneously for all n, revisiting a reflection inequality by James, which is of independent interest for the study of sequential decision making strategies.Lire moins >
Langue :
Anglais
Vulgarisation :
Non
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