The Annals of Statistics

Distributions Related to Linear Bounds for the Empirical Distribution Function

Jon A. Wellner

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$X_1, \cdots, X_n$ 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 $\Gamma_n^{-1}$.

Article information

Ann. Statist., Volume 5, Number 5 (1977), 1003-1016.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62E15: Exact distribution theory
Secondary: 62G30: Order statistics; empirical distribution functions 60F05: Central limit and other weak theorems

Distributions linear bounds empirical distribution function Poisson process


Wellner, Jon A. Distributions Related to Linear Bounds for the Empirical Distribution Function. Ann. Statist. 5 (1977), no. 5, 1003--1016. doi:10.1214/aos/1176343955.

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