Atomic functions and nonparametric estimates of the probability density

Nonparametric estimates of the probability density and its derivatives are considered on the basis of the theory of atomic functions (AF). New constructions of weight functions (WF) with compact support are proposed and substantiated that allow building admissible estimates of both the probability density in itself and its first and second derivatives. The distribution of a series of s-dimensional random variables is considered to converge to the s-dimensional normal distribution with the vector of means and the covariance matrix. To study spectral kernels, the modified physical characteristics, such as the width of the spectral density function, the relative width of the spectral density function, and the maximum level of side lobes are considered. The admissible nonparametric estimations of the probability density and its first and second derivatives for the sequence of random variables are constructed.