Limit theorems for the ratio of the empirical distribution function to the true distribution function期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Limit theorems for the ratio of the empirical distribution function to the true distribution function

Authors:	Jon A Wellner

Institution:	(1) Department of Statistics, University of Rochester, 14627 Rochester, New York, USA

Abstract:	Summary We consider almost sure limit theorems for $\begin{gathered} \parallel \Gamma _n /I\parallel _{ a_{ n} }^{ 1} \equiv \sup (\Gamma _n (t)/t) \hfill \\ a_{ n} \leqq t \leqq 1 \hfill \\ \end{gathered}$ and $\begin{gathered} \parallel I/\Gamma _n \parallel _{ a_{ n} }^{ 1} = \sup (t/\Gamma _n (t)) \hfill \\ a_{ n} \leqq t \leqq 1 \hfill \\ \end{gathered}$ where _n is the empirical distribution function of a random sample ofn uniform (0, 1) random variables anda _n0. It is shown that (1) ifna _n/log₂ n then both $\parallel \Gamma _n /I\parallel _{ a_{ n} }^{ 1}$ and $\parallel I/\Gamma _n \parallel _{ a_{ n} }^{ 1}$ converge to 1 a.s.; (2) ifna _n/log₂ n=d>0 (d>1) then $\parallel \Gamma _n /I\parallel _{ a_{ n} }^{ 1} ( \parallel I/\Gamma _n \parallel _{ a_{ n} }^{ 1} )$ has an almost surely finite limit superior which is the solution of a certain transcendental equation; and (3) ifna _n/log₂ n0 then $\parallel \Gamma _n /I\parallel _{ a_{ n} }^{ 1}$ and $\parallel I/\Gamma _n \parallel _{ a_{ n} }^{ 1}$ have limit superior + almost surely. Similar results are established for the inverse function _n ^–1 .Supported by the National Science Foundation under MCS 77-02255

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