On Exact Distribution of a Class of Supremum-type\\Statistics for Goodness of Fit

For goodness of fit tests with simple null hypothesis, Zhang (2002) constructed a classes of supremum-type tests. Different parameter \lambda and different weighted function q(t) result in different tests, including the Kolmogorov-Smirov test, Berk and Jones (1979) test and so on. So far, only a few tests corresponding to particular \lambda and q(t) have been studied in the literature. However, for different problems, the ``best'' tests are different. It is necessary to discuss the tests for all \lambda and the general q(t). In this paper, the exact distributions of the test statistics for all \lambda and q(t)\equiv 1 are derived. When sample size n is large, it takes a long time to get the exact quantile. So we give some advice on the computation methods for different sample size by simulation studies, and a real example to simply illustrate the above methods.