Using the inverse Pearson’s chi-square test in the multiplicative synthesis of new statistical tests from already known tests to test the hypothesis of normal distribution of small sample data

Abstract

Background. The study considers the issue of analyzing small samples by synthesizing new statistical tests generated by combining the classical statistical chi-square Pearson test and other well-known statistical tests. Methods. It is proposed to perform the inversion of the chi-square test by shifting it, scaling and dividing one by its final result. It is proposed to obtain new statistical criteria by multiplying the inverse Pearson’s chi-square test by the results of convolutions of small samples according to such classical criteria as the Smirnov-Cramer-von Mises test and the Anderson-Darling test. Results and conclusions. For the product of the inverse chi-square test and the Smirnov-Kramer-von Mises test, it is possible to reduce the probabilities of errors of the first and second kind by more than 1.45 times. By analogy with the chi-square test, inverse statistical tests can be obtained for any currently known statistical tests for testing the hypothesis of normality of small samples, which opens up the possibility of obtaining many new statistical tests by their multiplicative combination in pairs, triplets and other groups.

Keywords

• statistical analysis of small samples
• testing of the hypothesis of normality
• pearson’s chi-square test
• smirnov-kramer-von mises test
• anderson-darling test