Comparison of Optimization Methods for the Regression Estimation of the Probability Density of a One-Dimensional Random Variable

The methods of selecting the bandwidth of kernel functions for the regression estimation of the probability density of a one-dimensional random variable are studied. The regression estimation of the probability density is a modification of Rosenblatt–Parzen statistics and used in big statistical data processing. Its synthesis is based on the compression of an initial sample by decomposing the range of random variable values. Resulting data array elements are the centers of sampling intervals and the frequencies of belonging to them for random variable values from the initial sample. This information is sufficient to estimate the probability density of a random variable in the form of nonparametric regression. Therefore, it becomes possible to select the bandwidth for the kernel functions of regression estimation from the condition of its minimum approximation error for the desired probability density. The traditional approach of nonparametric estimation optimization to the probability density is based on the minimization of its mean-square deviation. The approximation properties of regression estimation for the probability density are analyzed by using the considered methods of its optimization.