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September, 1977 Distributions Related to Linear Bounds for the Empirical Distribution Function
Jon A. Wellner
Ann. Statist. 5(5): 1003-1016 (September, 1977). DOI: 10.1214/aos/1176343955


$X_1, \cdots, X_n$ are i.i.d. Uniform (0, 1) rv's with empirical df $\Gamma_n$ and order statistics $0 < U_1 < \cdots < U_n < 1.$ Define random variables $U_\ast, i_\ast$ (for $n \geqq 2$) by $\max_{1\leqq i \leqq n - 1} \frac{U_{i + 1}}{i} = \frac{U_{i_\ast} + 1}{i_\ast}, U_\ast = U_{i_\ast + 1};$ $i_\ast + 1$ is the (random) index of the order statistic at which the maximum is achieved and $U_\ast$ is the value of that order statistic. The distributions of $(U_\ast, i_\ast)$ and of $U_\ast$ and $i_\ast$ are found for all $n \geqq 2,$ extending and complementing earlier results due to Birnbaum and Pyke, Chang, and Dempster. The limiting distributions are found and related to the corresponding sums of exponential rv's by a Poisson type invariance result for the empirical df $\Gamma_n$ and its inverse $\Gamma_n^{-1}$.


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Jon A. Wellner. "Distributions Related to Linear Bounds for the Empirical Distribution Function." Ann. Statist. 5 (5) 1003 - 1016, September, 1977.


Published: September, 1977
First available in Project Euclid: 12 April 2007

zbMATH: 0368.62027
MathSciNet: MR458673
Digital Object Identifier: 10.1214/aos/1176343955

Primary: 62E15
Secondary: 60F05 , 62G30

Keywords: distributions , Empirical distribution function , linear bounds , Poisson process

Rights: Copyright © 1977 Institute of Mathematical Statistics


Vol.5 • No. 5 • September, 1977
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