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

Abstract

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.

Key Words:

• Anderson-Darling test
• composite hypotheses testing
• Cramer-Mises-Smirnov test
• goodness-of-fit test
• Kolmogorov test

Additional information

Notes on contributors

Boris Yu. Lemeshko

Boris Yu. Lemeshko is Doctor of technical sciences (1997), Professor, Dean of Faculty of Applied Mathematics and Computer Science (Novosibirsk State Technical University, Russia). Scientific interests lie in the area of computer methods of data analysis and statistical regularities research in the failure of classical assumptions.

Stanislav B. Lemeshko

Stanislav B. Lemeshko is the scientific researcher of the Applied Mathematics Department (Novosibirsk State Technical University, Russia), Ph.D. (2007). Scientific interests are computer methods of research statistical regularities.