Dimension-Free Anticoncentration Bounds for Gaussian Order Statistics with Discussion of Applications to Multiple Testing

TL;DR

This paper establishes a dimension-free anticoncentration bound for Gaussian order statistics, which has significant implications for error rate control in high-dimensional multiple testing scenarios.

Contribution

It provides a novel, dimension-free anticoncentration inequality for Gaussian order statistics and discusses its applications in multiple hypothesis testing.

Findings

Bound on probability of Gaussian order statistic within an interval

Implications for error rate control in high-dimensional testing

Dimension-free nature of the anticoncentration bound

Abstract

The following anticoncentration property is proved. The probability that the $k$-order statistic of an arbitrarily correlated jointly Gaussian random vector $X$ with unit variance components lies within an interval of length $\varepsilon$ is bounded above by $2{\varepsilon}k ({ 1+\mathrm{E}[\|X\|_\infty ]}) $. This bound has implications for generalized error rate control in statistical high-dimensional multiple hypothesis testing problems, which are discussed subsequently.