On the study of the properties of estimates and the influence of estimation methods on the distribution of statistics of hypothesis testing criteria

Statistical expressions of many criteria for of testing statistical hypotheses contain estimates of parameters or numerical characteristics such as mathematical expectation or dispersion, of assumed laws of probability distribution. Estimates can be found by different methods and differ in properties. The study of asymptotic properties of some estimate by analytical methods is, as a rule, a non-trivial task. It is even more difficult to study the properties of estimates by analytical methods with limited sample sizes, when the real properties of estimates may differ significantly from the asymptotic ones. Computer technologies make it possible to study the asymptotic properties of estimates, to clarify at what sample sizes one can rely on asymptotic properties, and to show how real properties differ from asymptotic ones at limited sample sizes. As a consequence, the distributions of the statistics of the tests for the validity of the hypothesis being tested depend on the properties of the estimates used, that is, on the evaluation method used. The study of changed properties of tests due to the choice of another evaluation method using analytical methods also turns out to be a non-trivial task. At the same time, computer technologies make it possible not only to solve such problems, but also to ensure the correct application of relevant tests in changing conditions. Within the framework of the developed software system, the capabilities of computer technologies for modeling and data analysis in studying the properties of various assessments and the influence of the type of assessment on the distribution of statistic of criteria are demonstrated. The properties of MD estimates that minimize the statistics of various nonparametric goodness-of-fit tests are investigated. It is shown that, unlike maximum likelihood estimators, these estimators are not asymptotically efficient and their asymptotic distributions differ. It is shown that when testing complex hypotheses, when the parameters of the law are found from the analyzed sample, the distributions of the statistics of all nonparametric goodness-of-fit tests and the Pearson test, which occur when the hypothesis being tested is valid, depend on the method of estimating the parameters. Moreover, the distributions of statistics of the same test differ for different evaluation methods. The obtained estimates of the power of the criteria, knowledge of the asymptotic properties of the estimates and the nature of the influence of these estimates on the distribution of the statistics of the test allow us to judge the preference of the combinations of the applied tests with the evaluation methods. The software system allows conducting similar studies with respect to any parameters of more than 30 parametric models of distribution laws included in the system, and with respect to any of more than a dozen goodness-of-fit tests.