Synthesis of new, more powerful statistical tests through multiplicative clustering of classical Frozini and Murota-Takeuchi tests with the Hurst test for the purpose of testing small samples for normality

Abstract

Aim. The paper examines the problem of small sample analysis by means of synthesizing new statistical tests generated by the clustering of the Hurst statistical test with the Frozini test, as well as with the Murota-Takeuchi test. The problem of normal distribution hypothesis testing on samples of 16 to 25 experiments is solved. Such significant limitations of the sample size arise in subject areas that include biometrics, biology, medical science and economics. In this case, the problem can be solved by applying not one, but a number of statistical tests to the analysis of the same small sample. Methods. It is suggested multiplying the Hurst test outputs by the Frozini test and/or the Murota-Takeuchi test outputs. A multiplicative clustering was performed for pairs of examined tests and their combination. It was shown that for each known statistical test, an equivalent artificial neuron can be constructed. A neural network integration of about 21 classical statistical tests constructed in the last century becomes possible. It is expected that the addition of new statistical tests in the form of artificial neurons will improve the quality of multi-criteria analysis solutions. Formally, the products of non-recurrent pairs of 21 original classical statistical tests should produce 210 new statistical tests. That is significantly more than the total number of statistical tests developed in the last century for the purpose of normality testing. Results. Pairwise product of the examined tests allows reducing the probability of errors of the first and second kind by more than 1.55 times as compared to the basic Hurst test. In case of triple product of the tests, the probabilities of error decrease relative to the basic Hurst test and to the associated second test. It is noted that there is no steady improvement in the quality of the decisions made by multiplicative mathematical constructions. The probabilities of error of the new test obtained by multiplying three of the examined tests are approximately 1.5% worse as compared to those of the tests obtained by multiplying pairs of the original tests. Conclusions. By analogy with the examined tests, the proposed data processing methods can also be applied to other known statistical tests. In theory, it becomes possible to significantly increase the number of new statistical tests by multiplying their final values. Unfortunately, as the number of clustered statistical tests grows, mutual correlations between the newly synthesized tests grow as well. The latter fact limits the capabilities of the method proposed in the paper. Further research is required in order to identify the most efficient combinations of pairs, triples or large groups for known statistical tests.