Local Dvoretzky-Kiefer-Wolfowitz confidence bands

TL;DR

This paper develops precise local concentration inequalities for the supremum of empirical CDF deviations over sub-intervals, extending classical results and providing tools for risk measure estimation and sequential decision strategies.

Contribution

It derives exact expressions for local supremum probabilities of empirical CDF deviations, extending classical bounds to sub-intervals and uniform over sample sizes.

Findings

Exact formulas for local supremum probabilities

Comparison with classical DKW bounds

Extension to uniform concentration inequalities over all sample sizes

Abstract

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 two seminal papers. We focus on the concentration of the \emph{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 $P(\sup_{u\in[\underline{u},\overline{u}]}U_n(u)-U(u)>\epsilon)$ for different values of $\underline{u},\overline{u}\in[0,1]$. Such local controls naturally appear for instance when studying estimation error of spectral risk-measures (such as the conditional value at risk), where $[\underline{u},\overline{u}]$ is typically $[0,\alpha]$ or $[1-\alpha,1]$ for a risk level $\alpha$, 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 $P(\sup_{u\in[\underline{u},\overline{u}]}U_n(u)-U(u)>\epsilon)$ and $P(\sup_{u\in [\underline{u},\overline{u}]}U(u)-U_n(u)>\epsilon)$ for each $n,\epsilon,\underline{u},\overline{u}$. Interestingly these quantities, seen as a function of $\epsilon$, can be easily inverted numerically into functions of the probability level $\delta$. Although not explicit, they can be computed and tabulated. We plot such expressions and compare them to the classical bound $\sqrt{\frac{\ln(1/\delta)}{2n}}$ provided by Massart inequality. 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.