Stein鈥檚 method for nonlinear statistics: A brief survey and recent progress

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• 作者：Qi-Man Shao^a ; ^{qmshao@cuhk.edu.hk" class="auth_mail" title="E-mail the corresponding author} ; Kan Zhang^b ; Wen-Xin Zhou^c
• 关键词：Berry&ndash ; Esseen bound ; Cramé ; r-type moderate deviation ; Stein&rsquo ; s method ; Nonlinear statistics ; UU-statistics ; Studentized statistics
• 刊名：Journal of Statistical Planning and Inference
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：168
• 期：Complete
• 页码：68-89
• 全文大小：505 K

文摘

Stein’s method is a powerful tool for proving central limit theorems along with explicit error bounds in probability theory, where uniform and non-uniform Berry–Esseen bounds spark general interest. Nonlinear statistics, typified by Hoeffding’s class of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0378375815001342&_mathId=si1.gif&_user=111111111&_pii=S0378375815001342&_rdoc=1&_issn=03783758&md5=864b0cfb279bc8f56eef2057b6d0b502" title="Click to view the MathML source">Uclass="mathContainer hidden">class="mathCode">$U$-statistics, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0378375815001342&_mathId=si2.gif&_user=111111111&_pii=S0378375815001342&_rdoc=1&_issn=03783758&md5=d356d570338d26ea9dfba72af0d59825" title="Click to view the MathML source">Lclass="mathContainer hidden">class="mathCode">$L$-statistics, random sums and functions of nonlinear statistics, are building blocks in various statistical inference problems. However, because the standardized statistics often involve unknown nuisance parameters, the Studentized analogues are most commonly used in practice. This paper begins with a brief review of some standard techniques in Stein’s method, and their applications in deriving Berry–Esseen bounds and Cramér moderate deviations for nonlinear statistics, and then using the concentration inequality approach, establishes Berry–Esseen bounds for Studentized nonlinear statistics in a general framework. As direct applications, sharp Berry–Esseen bounds for Studentized class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0378375815001342&_mathId=si1.gif&_user=111111111&_pii=S0378375815001342&_rdoc=1&_issn=03783758&md5=864b0cfb279bc8f56eef2057b6d0b502" title="Click to view the MathML source">Uclass="mathContainer hidden">class="mathCode">$U$- and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0378375815001342&_mathId=si2.gif&_user=111111111&_pii=S0378375815001342&_rdoc=1&_issn=03783758&md5=d356d570338d26ea9dfba72af0d59825" title="Click to view the MathML source">Lclass="mathContainer hidden">class="mathCode">$L$- statistics are obtained.