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
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.
Ann. Statist., Volume 7, Number 1 (1979), 108-115.
First available in Project Euclid: 12 April 2007
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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
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