Analysis of optimization methods for nonparametric estimation of probability density in large samples

The article proposes a procedure for selecting the blur factor of kernel functions for nonparametric density estimation of a one-dimensional random variable given large amounts of statistical data, for example, obtained via the remote sensing of natural objects. The proposed procedure uses regression density estimation. A procedure is presented for synthesizing a regression density estimate. The estimate synthesis is based on the original sample compression by decomposing the range of a random variable. To this end, the Heinhold-Gaede rule and a formula for choosing the optimal number of sampling intervals are applied. The article considers two approaches to selecting the blur factor of regression density estimation using the conventional method and that proposed by the authors to optimize nonparametric density estimation. The conventional method for optimizing nonparametric density estimation is based on minimizing its standard deviation. In the proposed method, the choice of the blur factors of kernel functions relies on the conditions for the minimum approximation error of regression density estimation. The article analyzes the approximation properties of regression density estimation using two optimization methods. The conditions of their competence in estimating the probability densities of random variables following lognormal distribution are established. The results obtained for a one-dimensional random variable can be used to optimize the regression density estimation of a multi-dimensional random variable.