## The Annals of Statistics

- Ann. Statist.
- Volume 7, Number 1 (1979), 108-115.

### The Asymptotic Distribution of the Supremum of the Standardized Empirical Distribution Function on Subintervals

#### Abstract

It is well known that the limit distribution of the supremum of the empirical distribution function $F_n$ centered at its expectation $F$ and standardized by division by its standard deviation is degenerate, if the supremum is taken on too large regions $\varepsilon_n < F(u) < \delta_n$. So it is natural to look for sequences of linear transformations, so that for given sequences of sup-regions $(\varepsilon_n, \delta_n)$ the limit of the transformed sup-statistics is nondegenerate. In this paper a partial answer is given to this problem, including the case $\varepsilon_n \equiv 0, \delta_n \equiv 1$. The results are also valid for the Studentized version of the above statistic, and the corresponding two-sided statistics are treated, too.

#### Article information

**Source**

Ann. Statist., Volume 7, Number 1 (1979), 108-115.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176344558

**Digital Object Identifier**

doi:10.1214/aos/1176344558

**Mathematical Reviews number (MathSciNet)**

MR515687

**Zentralblatt MATH identifier**

0398.62013

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62E20: Asymptotic distribution theory

Secondary: 60F05: Central limit and other weak theorems

**Keywords**

Standardized empirical distribution function normalized sample quantile process extreme value distribution boundary crossing of empirical process Poisson process Ornstein-Uhlenbeck process normalized Brownian bridge process goodness of fit test tail estimation

#### Citation

Jaeschke, D. The Asymptotic Distribution of the Supremum of the Standardized Empirical Distribution Function on Subintervals. Ann. Statist. 7 (1979), no. 1, 108--115. doi:10.1214/aos/1176344558. https://projecteuclid.org/euclid.aos/1176344558