On a variance dependent Dvoretzky-Kiefer-Wolfowitz inequality

TL;DR

This paper establishes a variance-dependent Dvoretzky-Kiefer-Wolfowitz inequality that provides sharp bounds on the empirical distribution function's deviation from the true distribution, depending on the variance at each point.

Contribution

The paper introduces a new inequality that refines the DKW bound by incorporating the variance of the distribution, achieving near-optimal bounds.

Findings

The inequality holds with high probability for all points with distribution function values in [Δ, 1-Δ].

The bounds depend on the local variance, improving upon classical uniform bounds.

The results are shown to be nearly optimal up to constants.

Abstract

Let $X$ be a real-valued random variable with distribution function $F$. Set $X_1,\dots, X_m$ to be independent copies of $X$ and let $F_m$ be the corresponding empirical distribution function. We show that there are absolute constants $c_0$ and $c_1$ such that if $\Delta \geq c_0\frac{\log\log m}{m}$, then with probability at least $1-2\exp(-c_1\Delta m)$, for every $t\in\mathbb{R}$ that satisfies $F(t)\in[\Delta,1-\Delta]$, \[ |F_m(t) - F(t) | \leq \sqrt{\Delta \min\{F(t),1-F(t)\} } .\] Moreover, this estimate is optimal up to the multiplicative constants $c_0$ and $c_1$.