Dependence Between Histogram Parameters and the Kernel Estimate of a Unimodal Probability Density

The dependence between the sampling interval of the domain of values of a one-dimensional random variable and the blur coefficient of the kernel probability density estimate is determined. The studies used the results of an analysis of the asymptotic properties of a nonparametric estimate of the probability density of the Rosenblatt–Parzen type and its modification. It is shown that the modification of the kernel probability density estimate is a smoothed histogram. The optimal expressions for the kernel function blur coefficient and the length of the sampling interval of the domain of values of a one-dimensional random variable are considered. These parameters are obtained from the condition of minimum mean square deviations of the considered probability density estimates. On this basis, a relationship was established between the studied parameters, which is determined by a constant and depends on the applied kernel function and the volume of the initial statistical data. The values of the detected constant are characterized by the form of the reconstructed probability density and are independent of its parameters. According to the data of computational experiments, formulas are proposed for estimating the analyzed constant by the value of the antikurtosis coefficient for symmetric and asymmetric distribution laws. To estimate the antikurtosis coefficient, we used the initial statistical data in the problem of reconstructing the probability density. The results obtained make it possible to quickly determine the length of the sampling interval from the value of the kernel function blur coefficient, which is relevant when testing hypotheses about the distributions of random variables. The presented conclusions are confirmed by the results of computational experiments.