Concentration Inequalities for Statistical Inference (Communications in Mathematical Research, 2021)
This paper gives a review of concentration inequalities which arewidely employed in non-asymptotical analyses of mathematical statistics ina wide range of settings, from distribution-free to distribution-dependent, fromsub-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 ofhigh-dimensional linear and Poisson regressions are also provided. We aim toillustrate the concentration inequalities with known constants and to improveexisting bounds with sharper constants.
Keywords: Constants-specified inequalities; sub-Weibull random variables; heavy-tailed distributions; high-dimensional estimation and testing; finite-sample theory; random matrices.