Estimation of the Frocini criteria and omega square criteria statistics by the statistical tests method for a mixture of normal distributions

A lot of sets of subjects and objects in biology, industry, management can be divided into a number of classes, each of which corresponds to a certain distribution component. When analyzing a mixture of distributions, it is necessary to estimate its parameters (task 1) and to assess the correspondence of empirical and theoretical distribution functions (task 2). To solve the first problem, numerical algorithms that implement the method of moments and the maximum likelihood method are used. In this paper, the problem of estimating the distribution parameters is solved by minimizing the good- ness measure by the Quasi-Newton method. The second problem is solved by comparing the empirical and theoretical distribution functions by one or several statistical goodness measures. Statistics of the distribution of these measures depends on the sample size, the method of forming data and estimating distribution parameters. The paper examines the goodness measure between Frocini and omega-square (Kramer - Mises - Smirnov). The evaluation of the statistics of the goodness measure was carried out by the simulation method based on the results of 50000 statistical tests. In each of the tests, the distribution parameters were estimated by minimizing the calculated value of the corresponding goodness measure. The results of simulation modeling allow estimating the statistics of the parameters of a mixture of distributions. The results of solving the considered problems for a mixture of two normal distributions of size 240 are pre- sented.