Exact and asymptotic goodness-of-fit tests based on the maximum and its location of the empirical process. (English) Zbl 07692853
Summary: The supremum of the standardized empirical process is a promising statistic for testing whether the distribution function \(F\) of i.i.d. real random variables is either equal to a given distribution function \(F_0\) (hypothesis) or \(F\geq F_0\) (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-\(\alpha\) test. However, the rate of convergence is extremely slow. As a consequence the probability of the type I error is much larger than \(\alpha\) even for sample sizes beyond 10.000. Now, the standardization consists of the weight-function \(1/\sqrt{F_0 (x)(1-F_0 (x))}\). 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 distribution) under the hypothesis. A comparison via a Monte-Carlo simulation shows that the new test is uniformly better than the Smirnov-test and an appropriately modified test due to (The Annals of Statistics 11 (1983) 933-946). Our methodology also works for the two-sided alternative \(F\neq F_0\).
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