The Asymptotic Distribution of the Suprema of the Standardized Empirical Processes

Abstract

The supremum of the empirical distribution function $F_n$ centered at its expectation $F$ and standardized by division by its standard deviation has recently been shown by Jaeschke to have asymptotically an extreme-value distribution after a second location and scale transformation depending only on the sample size $n$. In this paper the studentized form of the above statistic, obtained by division by the estimated standard deviation, is shown to have the same large sample behavior. This statement is equivalent to the analogous assertion for the standardized sample quantile process for the uniform distribution. The three results imply each other. The present result yields immediately confidence regions that contract to zero width in the tails. The proofs given here rest on a limit theorem by Darling and Erdos on the maxima of standardized partial sums of i.i.d. random variables. In addition, Kolmogorov's theorem is used.