2022/07/16 by Ferger, Dietmar · 1 citation
#FOS: Mathematics #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.2207.07947
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 F0 (hypothesis) or F ≥ F0 (one-sided alternative). Since \citer5 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 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 \citer20. Our methodology also works for the two-sided alternative F ≠ F0.