The paper studies the problem associated with the optimization of nonparametric probability density estimates, whose relevance is attributed to the lower efficiency of nonparametric algorithms for data processing with the increasing amount of statistical data. In this study, the authors examine a procedure for optimizing the kernel density estimation of a two-dimensional random variable having independent components. The possibility of using the optimal bandwidths of the kernel density estimates of one-dimensional random variables when synthesizing the two-dimensional nonparametric probability density of a random variable having independent components is justified. The proposed approach relies on the asymptotic properties of Rosenblatt–Parzen nonparametric probability density estimation. For a two-dimensional random variable, it is shown that the main contribution to the asymptotic expression for standard deviation is made by the corresponding criteria for one-dimensional random variables. When estimating two-dimensional probability density, it is possible to use bandwidths to minimize the standard deviations of one-dimensional random variables. The obtained conclusions are confirmed by the results of computational experiments in the analysis of normal distribution laws. The possibility of developing the proposed procedure for optimizing the nonparametric probability density estimates of multidimensional random variables having independent components is demonstrated.