Estimation of the Parameters of Fractional-Stable Laws by the Method of Minimum Distance*

Abstract

Algorithm for statistical estimation of the parameters of fractional-stable distributions is described in this article. This algorithm is constructed on the basis of the method of distance minimization between the empirical and theoretical distributions. As the distance between the two distributions, the χ distance is considered. The main difficulty in dealing with fractional-stable distributions is the absence of explicit expressions for the probability density function. That is why the theoretical density is estimated by the histogram method. The results of the test calculations and the results of the estimation of the quadratic deviation are presented. The results obtained by this estimator are compared with the results obtained by the method of moments. An example of the use of the estimator for the approximation of the experimental data obtained in the investigation of gene expression by RNA sequencing technology is given as well. It is shown that the probability density function of the gene expression in a wide-enough domain can be described by fractional-stable distributions.