{"title":"Uniform Strong Consistency of Rao-Blackwell Distribution Function Estimators","authors":"F. O'Reilly, C. Quesenberry","doi":"10.1214/AOMS/1177692401","DOIUrl":null,"url":null,"abstract":"In the independent sampling model, Rao-Blackwell distribution function estimators $\\tilde{F}_n(x)$ obtained by conditioning on sufficient statistics $T_n(X_1, \\cdots, X_n)$ are considered. If for each $n \\geqq 1, T_n$ is symmetric in $X_1,\\cdots, X_n$ and $T_{n+1}$ is $\\mathscr{B}(T_n, X_{n+1})$ measurable, it is shown that $\\tilde{F}_n(x)$ converges strongly to the corresponding $F(x)$ and uniformly in $x$. This is a direct generalization of the Glivenko-Cantelli theorem.","PeriodicalId":50764,"journal":{"name":"Annals of Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"1972-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Mathematical Statistics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1214/AOMS/1177692401","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 6
Abstract
In the independent sampling model, Rao-Blackwell distribution function estimators $\tilde{F}_n(x)$ obtained by conditioning on sufficient statistics $T_n(X_1, \cdots, X_n)$ are considered. If for each $n \geqq 1, T_n$ is symmetric in $X_1,\cdots, X_n$ and $T_{n+1}$ is $\mathscr{B}(T_n, X_{n+1})$ measurable, it is shown that $\tilde{F}_n(x)$ converges strongly to the corresponding $F(x)$ and uniformly in $x$. This is a direct generalization of the Glivenko-Cantelli theorem.