Modified Fast Algorithm for the Bandwidth Selection of the Kernel Density Estimation

A modification of the fast algorithm for the bandwidth selection of kernel functions in a nonparametric probability density estimate of the Rosenblatt-Parzen type is proposed. Fast algorithms for optimizing kernel estimates of probability densities make it possible to significantly reduce the calculation time when selecting their smoothing parameters (bandwidths) in comparison with the traditional approach, which is especially important when processing large statistical data. The method is based on the analysis of the formula for the optimal calculation of the smoothing parameter of kernel functions and the discovered dependence between the nonlinear functional on the second derivative of the reconstructed probability density and the antikurtosis coefficient. The proposed algorithm for the bandwidth selection provides a decrease in the probability density approximation error in comparison with the traditional approach. The findings are confirmed by the results of computational experiments. Special attention is paid to the dependence of these properties on the amount of initial data.