The Annals of Statistics

Distributions Related to Linear Bounds for the Empirical Distribution Function

Jon A. Wellner

Full-text: Open access

Abstract

$X_1, \cdots, X_n$ 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

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

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176343955

Digital Object Identifier
doi:10.1214/aos/1176343955

Mathematical Reviews number (MathSciNet)
MR458673

Zentralblatt MATH identifier
0368.62027

JSTOR
links.jstor.org

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

Keywords
Distributions linear bounds empirical distribution function Poisson process

Citation

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. https://projecteuclid.org/euclid.aos/1176343955


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