Exact and asymptotic goodness-of-fit tests based on the maximum and its location of the empirical process
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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 ≥ F_0 (one-sided alternative). Since <cit.> 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/√(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 <cit.>. Our methodology also works for the two-sided alternative F ≠ F_0.
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