## The Annals of Statistics

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

### Distributions Related to Linear Bounds for the Empirical Distribution Function

#### 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