Problems of nonparametric goodness-of-fit test application in tasks of measurement results processing

It is argued that in most cases two reasons underlie the incorrect application of nonparametric goodness-of-fit tests in various applications. The first reason is that when testing composite hypotheses and evaluating the parameters of the law for the analyzed sample, classical results associated with testing simple hypotheses are used. When testing composite hypotheses, the distributions of goodness-of-fit statistics are influenced by the form of the observed law F(x, q) corresponding to the hypothesis being tested, by the type and number of estimated parameters, by the estimation method, and in some cases by the value of the shape parameter. The paper shows the influence of all mentiomed factors on the distribution of test statistics. It is emphasized that, when testing composite hypotheses, the neglect, of the fact that the test has lost the property of “freedom from distribution” leads to an increase in the probability of the 2nd kind errors. It is shown that the distribution of the statistics of the test necessary for the formation of a conclusion about the results of testing a composite hypothesis can be found using simulation in an interactive mode directly in the process of testing. The second reason is associated with the presence of round-off errors which can significantly change the distributions of test statistics. The paper shows that asymptotic results when testing simple and composite hypotheses can be used with round -off errors D much less than the standard deviation s of the distribution law of measurement errors and sample sizes n not exceeding some maximum values. For sample sizes larger than these maximum values, the real distributions of the test statistics deviate from asymptotic ones towards larger statistics values. In such situations, the use of asymptotic distributions to arrive at a conclusion about the test results leads to an increase in the probabilities of errors of the 1st kind (to the rejection of a valid hypothesis being tested). It is shown that when the round-off errors and s are commensurable, the distributions of the test statistics deviate from the asymptotic distributions for small n. And as n grows, the situation only gets worse. In the paper, changes in the distributions of statistics under the influence of rounding are demonstrated both when testing both simple and composite hypotheses. It is shown that the only way out that ensures the correctness of conclusions according to the applied tests in such non-standard conditions is the use of real distributions of statistics. This task can be solved interactively (in the process of verification) and rely on computer research technologies and the apparatus of mathematical statistics.