TY - JOUR T1 - Concentration Inequalities for Statistical Inference AU - Zhang , Huiming AU - Chen , Songxi JO - Communications in Mathematical Research VL - 1 SP - 1 EP - 85 PY - 2021 DA - 2021/02 SN - 37 DO - http://doi.org/10.4208/cmr.2020-0041 UR - https://global-sci.org/intro/article_detail/cmr/18625.html KW - Constants-specified inequalities, sub-Weibull random variables, heavy-tailed distributions, high-dimensional estimation and testing, finite-sample theory, random matrices. AB -

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.