Concentration Inequalities for Statistical Inference

TL;DR

This paper reviews concentration inequalities used in statistical inference, covering various distributions and settings, and introduces new results with sharper constants, especially relevant for high-dimensional data analysis.

Contribution

It provides a comprehensive review of concentration inequalities with new results and sharper bounds, including applications to high-dimensional regression models.

Findings

New concentration bounds with sharper constants

Extensions to high-dimensional linear and Poisson regressions

Improved understanding of distribution-dependent inequalities

Abstract

This paper gives a review of concentration inequalities which are widely employed in non-asymptotical analyses of mathematical statistics in a wide range of settings, from distribution-free to distribution-dependent, from sub-Gaussian to sub-exponential, sub-Gamma, and sub-Weibull random variables, and from the mean to the maximum concentration. This review provides results in these settings with some fresh new results. Given the increasing popularity of high-dimensional data and inference, results in the context of high-dimensional linear and Poisson regressions are also provided. We aim to illustrate the concentration inequalities with known constants and to improve existing bounds with sharper constants.