On the properties and problems of applying goodness-of-fit tests based on the use of entropy and Kullback – Leibler divergence estimates

To verify the adequacy of the constructed models of distribution laws of random variables, various nonparametric goodness-of-fit tests are usually used, in particular, Kolmogorov, Kramer – Mises – Smirnov, Anderson – Darling, Kuiper, and Watson. When a simple hypothesis is tested, nonparametric goodness-of-fit tests are “distribution-free”: the asymptotic distributions of statistics do not depend on the type of law against which the hypothesis is tested. When testing composite hypotheses, when the parameters of the assumed law are estimated from a sample, the property of “freedom from distribution” is lost, and the distributions of statistics become dependent on a number of factors. In such situations, the use of nonparametric goodness-of-fit tests is possible only with the support of appropriate software that allows the achieved significance level Pv to be assessed using simulation modeling. The distributions of the statistics of the Zhang tests, which are a development of the Kolmogorov, Kramer – Mises – Smirnov, and Anderson – Darling tests, respectively, depend on the sample sizes, so their wide application in testing simple and complex hypotheses is possible only with the support of the Monte Carlo method. Distributions of goodness-of-fit tests statistics (when testing simple and composite hypotheses) can vary significantly due to the natural presence of rounding errors. A signal about the possibility of such a situation is the presence of a significant number of repeating values in the analyzed samples. In such situations, making a decision on the results of the inspection is also impossible without the use of simulation modeling. In recent years, several criteria have been proposed, aimed, for example, at checking whether samples belong to a normal or uniform law, the statistics of which are based on various entropy estimates. As experience shows, with respect to some competing hypotheses, such criteria demonstrate higher power estimates compared to classical nonparametric goodness-of-fit tests. When constructing the statistics of the Noughabi test to distinguish between two hypotheses, the Kullback-Leibler divergence was used, and the estimate proposed by Vasicek was taken as an estimate of entropy. This paper shows how the distributions of the Noughabi test statistics depend on the sample size n and the window size m, and how the distributions of the test statistics change when testing various composite hypotheses. The power of the criterion in testing norma-lity against various competing hypotheses was investigated. It is shown how, for given n, the power depends on the size of the “window” m. The existence of an optimal m is shown, at which the power is maximum relative to the competing hypothesis under consideration. It is shown that for a given n, the optimal values of m, as a rule, do not coincide for different competing hypotheses. Obviously, the application of such criteria in practice also implies the use of appropriate software and simulation modeling.