Application of criteria for testing homogeneity of distribution laws

The necessity of general homogeneity hypothesis checking, i.e. whether two (or more) samples of random variables belong to the same general aggregate, arises constantly in the course of analysis of unbiased errors of measuring instruments. Such task arises naturally in the course of measuring instruments calibration and when comparing lab tests results. Technologists, medical researchers and biologists also face the same task when processing results of experimental research. The present paper deals with the Smirnov, the Lehmann-Rosenblatt and the Anderson-Darling two-samples homogeneity tests; statistics for these tests is given; advantages and disadvantages of the tests are discussed. As a drawback of the Smirnov test we may mention a substantial discreteness of statistics distribution which must be taken into account when having significant values of n and equal number of m and n samples. This drawback can be overcome if we choose mutually prime integers as m and n . But even when this drawback has been remedied the real statistics distribution still differs substantially from the Kolmogorov limit distribution as the checked hypothesis is true. Hence when using the latter for evaluation of the significance level (p_value) incorrect conclusions can be reached. One can avoid this only by modifying the test statistics. As opposed to the Smirnov test, the Lehmann-Rosenblatt and the Anderson-Darling statistics distribution homogeneity tests do not actually differ from their limit distributions when having samples values of m,n > 25. Comparative analysis of tests powers under discussion, conducted with the help of statistic modeling methods, showed that, as a rule, the Anderson-Darling test boasts of bigger power than the Lehmann-Rosenblatt test, especially in case of differences of samples in their measure of dispersion. At the same time, when having rather similar yet competing hypothesis and smaller number of samples, the Lehmann-Rosenblatt test can show advantage in power. The Smirnov test yields to the Lehmann-Rosenblatt and the Anderson-Darling competing tests, but in certain cases it can be quite competitive. Previously, information on distribution of statistics for к-sample Anderson-Darling homogeneity test has been available only within a limited table of critical limits. In the present case the study of statistics distribution through statistic modeling methods with actual к values showed the presence of corresponding limit distributions. Results of such modeling showed that when using this test we can disregard the difference between the statistics distributions and the corresponding limit values while the number of samples n_t > 30. Based on the results of statistics modeling, approximate models for limit distributions of the Anderson-Darling k-samples test for к = 2 +11. This paper shows the models created, represented by the laws of beta-distributions of the IlIrd with particular parameter values. Such models produced with the help of the Anderson-Darling к-sample test enable finding the values of p_value, thus making the results of statistics conclusions more informative and more substantiated. The possibility of using the Z hang к-sample homogeneity test with statistics of Z_K , Z_C and Z_A is discussed, these being the extension of the Smirnov, the Lehmann-Rosenblatt and the Anderson-Darling test, respectively. The Zhang tests have some advantages in power with reference to the scale alternatives, but they somewhat yield to the Smirnov, the Lehmann-Rosenblatt and the Anderson-Darling tests with reference to the shift alternatives.

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