Exact and asymptotic goodness-of-fit tests based on the maximum and its location of the empirical process

Publikation: Beitrag in FachzeitschriftForschungsartikelBeigetragenBegutachtung



The supremum of the standardized empirical process is a promis-ing statistic for testing whether the distribution function F of i.i.d. real random variables is either equal to a given distribution function F0 (hypothesis) or F ≥ F0 (one-sided alternative). Since (The Annals of Statistics 7 (1979) 108–115) it is well-known that an affine-linear transformation of the suprema converge in distribution to the Gumbel law as the sample size tends to infinity. This enables the construction of an asymptotic level-α test. However, the rate of convergence is extremely slow. As a consequence the probability of the type I error is much larger than α even for sample sizes beyond 10.000. Now, the standardization consists of the weight-function 1/ F0 (x)(1 − F0 (x)). Substituting the weight-function by a suitable random constant leads to a new test-statistic, for which we can derive the exact distribution (and the limit dis-tribution) under the hypothesis. A comparison via a Monte-Carlo simulation shows that the new test is uniformly better than the Smirnov-test and an ap-propriately modified test due to (The Annals of Statistics 11 (1983) 933–946). Our methodology also works for the two-sided alternative F ≠ F0 .


Seiten (von - bis)140-154
FachzeitschriftBrazilian journal of probability and statistics
PublikationsstatusVeröffentlicht - März 2023


ASJC Scopus Sachgebiete


  • empirical process, Goodness of fit, measurability and continuity of the argmax-functional