Concentration inequalities for order statistics

TL;DR

This paper develops non-asymptotic variance and tail bounds for order statistics, including an exponential Efron-Stein inequality, with applications to Gaussian samples, advancing understanding of concentration properties beyond classical inequalities.

Contribution

It introduces a new exponential Efron-Stein inequality for order statistics with non-decreasing hazard rates, providing tighter bounds for Gaussian samples and extending concentration inequality techniques.

Findings

Derived asymptotically tight bounds for order statistics.

Established an exponential Efron-Stein inequality for distributions with non-decreasing hazard rate.

Provided variance and tail bounds for Gaussian order statistics.

Abstract

This note describes non-asymptotic variance and tail bounds for order statistics of samples of independent identically distributed random variables. Those bounds are checked to be asymptotically tight when the sampling distribution belongs to a maximum domain of attraction. If the sampling distribution has non-decreasing hazard rate (this includes the Gaussian distribution), we derive an exponential Efron-Stein inequality for order statistics: an inequality connecting the logarithmic moment generating function of centered order statistics with exponential moments of Efron-Stein (jackknife) estimates of variance. We use this general connection to derive variance and tail bounds for order statistics of Gaussian sample. Those bounds are not within the scope of the Tsirelson-Ibragimov-Sudakov Gaussian concentration inequality. Proofs are elementary and combine R\'enyi's representation of order statistics and the so-called entropy approach to concentration inequalities popularized by M. Ledoux.