Optimization of the kernel probability density estimation of a two-dimensional random variable with independent components

Abstract

The problem of optimization of nonparametric estimates of probability densities is studied, the relevance of which is due to a decrease in the efficiency of nonparametric algorithms for information processing with an increase in the volume of statistical data. A method for optimizing the kernel probability density estimation of a two-dimensional random variable with independent components is considered. The possibility of using optimal bandwidths of kernel probability densities estimates of one-dimensional random variables in the synthesis of a nonparametric two-dimensional probability density of a random variable with independent components is substantiated. The proposed approach is based on the results of a study of the asymptotic properties of a nonparametric Rosenblatt-Parson probability density estimation. For a two-dimensional random variable, it is shown that the main contribution to the asymptotic expression of the mean square deviation is provided by the corresponding criteria for one-dimensional random variables. Therefore, it becomes possible to use the bandwidths that minimize the mean square deviations of one-dimensional random variables when estimating a two-dimensional probability density. The obtained conclusions are confirmed by the results of computational experiments in the analysis of normal distribution laws. The possibility of developing the proposed methodology is shown when optimizing nonparametric estimates of the probability densities of multidimensional random variables with independent components.