Local Dvoretzky-Kiefer-Wolfowitz confidence bands

Maillard, Odalric-Ambrym

Document type :

Article dans une revue scientifique

DOI :

10.3103/S1066530721010038

Title :

Local Dvoretzky-Kiefer-Wolfowitz confidence bands

Author(s) :

Maillard, Odalric-Ambrym [Auteur]

Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 [CRIStAL]
Inria Lille - Nord Europe

Journal title :

Mathematical Methods of Statistics

Publisher :

Allerton Press, Springer (link)

Publication date :

2022

ISSN :

1066-5307

English keyword(s) :

Cumulative Distribution Function
Concentration inequalities
DKW
Risk measure

HAL domain(s) :

Statistiques [stat]/Théorie [stat.TH]
Statistiques [stat]/Machine Learning [stat.ML]

English abstract : [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 ...
Show more >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.Show less >

Language :

Anglais

Peer reviewed article :

Oui

Audience :

Internationale

Popular science :

Non

Collections :

Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL) - UMR 9189

Source :

Harvested from HAL

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