Estimating the integral of the square of the probability density of a one-dimensional random variable

Abstract

When choosing the optimal number of sampling intervals of the range of values of a one-dimensional random variable, it is established that the functional of the square of the probability density is a constant. The values of the constant are independent of the probability density parameters. The functional dependencies of the studied constant on the coefficient of antikurtosis of the distribution law of a random variable are determined. The analysis of the established dependences for families of lognormal probability densities, Student's distribution laws and families of probability densities with Gauss distribution is carried out. Based on the results obtained, a generalized model is formed between the studied constant and the antikurtosis coefficient. The generalized model does not depend on the type of probability density, but is determined by the estimation of the antikurtosis coefficient. On this basis, we develop a method for estimating the integral of the square of the probability density, which involves the following actions. The random variable interval and the antikurtosis coefficient are estimated from the initial sample. At known values of these estimates, the integral of the square of the probability density is calculated. The effectiveness of the proposed method is confirmed by the results of computational experiments. The conditions of the computational experiment differ significantly from the information used in the synthesis of models of the dependence between the studied constant and the antikurtosis coefficient. The conditions of competence of the method of estimating the probability density square integral from the antikurtosis coefficient are established using the proposed models of the dependence of the studied constant on the conditions of the computational experiment.