Sharp Anti-Concentration Inequalities for Extremum Statistics via Copulas

TL;DR

This paper establishes sharp bounds on the concentration of the maximum of i.i.d. variables, introduces a new class of copulas for improved bounds, and applies these results to high-dimensional statistical inference.

Contribution

It provides the first dimension-independent anti-concentration bounds for maxima under mild dependence, introducing copulas with convex diagonal sections for sharper inequalities.

Findings

Sharp upper and lower bounds for maximum concentration functions.

Introduction of copulas with convex diagonal sections for better bounds.

Application of bounds to high-dimensional Gaussian mixture models.

Abstract

We derive sharp upper and lower bounds for the pointwise concentration function of the maximum statistic of $d$ identically distributed real-valued random variables. Our first main result places no restrictions either on the common marginal law of the samples or on the copula describing their joint distribution. We show that, in general, strictly sublinear dependence of the concentration function on the dimension $d$ is not possible. We then introduce a new class of copulas, namely those with a convex diagonal section, and demonstrate that restricting to this class yields a sharper upper bound on the concentration function. This allows us to establish several new dimension-independent and poly-logarithmic-in-$d$ anti-concentration inequalities for a variety of marginal distributions under mild dependence assumptions. Our theory improves upon the best known results in certain special cases. Applications to high-dimensional statistical inference are presented, including a specific example pertaining to Gaussian mixture approximations for factor models, for which our main results lead to superior distributional guarantees.