Local Dvoretzky–Kiefer–Wolfowitz Confidence Bands

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.