The Annals of Statistics

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

D. Jaeschke

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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


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