Construction of statistic Distribution Models for Nonparametric Goodness-of-Fit Tests in Testing Composite Hypotheses: The Computer Approach

In composite hypotheses testing, when the estimate of the scalar or vector parameter of the probabilities distribution laws is calculated by the same sample, the nonparametric goodness-of-fit Kolmogorov, Cramer-Mises-Smirnov, Anderson-Darling tests lose the free distribution property. In testing of composite hypotheses, the conditional distribution law of the statistic is affected by a number of factors: the form of the observed probabilities distribution law corresponding to the true testable hypothesis; the type of the parameter estimated and the number of parameters to be estimated; sometimes, it is a specific value of the parameter (e.g., in the case of gamma-distribution and beta-distribution families); the method of parameter estimation. In this paper we present more precise results (tables of percentage points and statistic distribution models) for the nonparametric goodness-of-fit tests in testing composite hypotheses using the maximum likelihood estimate (MSE) for some probabilities distribution laws. Statistic distributions of the nonparametric goodness-of-fit tests are investigated by the methods of statistical simulation. Constructed empirical statistic distributions are approximated with analytical law models.