## The Annals of Statistics

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

D. Jaeschke

#### 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

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