CHINESE JOURNAL OF APPLIED PROBABILITY AND STATIST 2009, 25(4) 409-420 DOI:      ISSN: 1001-4268 CN: 31-1256

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

Wei Bocheng:Li Guoying:Zhao Zhiyuan

College of Mathematical Sciences, Guangxi Normal University College of Applied Sciences, Beijing University of Technology Academy of Mathematics and System Sciences, Chinese Academy of Sciences

Abstract��

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.

Keywords�� Exact distribution   generalized nonparametric likelihood ratio   goodness-of-fit test.