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Volume 37, Issue 1
Concentration Inequalities for Statistical Inference

Huiming Zhang & Songxi Chen

Commun. Math. Res., 37 (2021), pp. 1-85.

Published online: 2021-02

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  • 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.

  • Keywords

Constants-specified inequalities, sub-Weibull random variables, heavy-tailed distributions, high-dimensional estimation and testing, finite-sample theory, random matrices.

  • AMS Subject Headings

60F10, 60G50, 62E17

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

zhanghuiming@pku.edu.cn (Huiming Zhang)

  • BibTex
  • RIS
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@Article{CMR-37-1, author = {Huiming and Zhang and zhanghuiming@pku.edu.cn and 14728 and School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China. and Huiming Zhang and Songxi and Chen and and 14727 and and Songxi Chen}, title = {Concentration Inequalities for Statistical Inference}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {37}, number = {1}, pages = {1--85}, 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.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2020-0041}, url = {http://global-sci.org/intro/article_detail/cmr/18625.html} }
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.

Huiming Zhang & Songxi Chen. (2021). Concentration Inequalities for Statistical Inference. Communications in Mathematical Research . 37 (1). 1-85. doi:10.4208/cmr.2020-0041
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