Estimation of a Nonlinear Functional of Probability Density when Optimizing Nonparametric Decision Functions

We propose a method for estimating the nonlinear functional of the probability density of a two-dimensional random variable. The technique applies to the implementation of procedures for the “quick” selection of blur coefficients in the problem of optimizing kernel estimates of the probability density and can significantly increase the computational efficiency of nonparametric decision rules. The proposed approach is based on the analysis of the formula for the optimal blur coefficient of the kernel probability density estimate. In this case, the kernel function blur coefficient is presented as a product of an indefinite parameter and standard deviations of random variables. The main component of the indefinite parameter is a nonlinear functional of the probability density, which for a family of unimodal symmetric distribution laws is determined by the form of the probability density and does not depend on the density parameters of. For a family of two-dimensional lognormal distribution laws of independent random variables, we determine the approximation errors of the considered nonlinear functional of the probability density. We study the possibility of using the proposed methodology for estimating nonlinear functionals on probability densities that differ from lognormal distribution laws. We analyze the influence of the occurring approximation errors on the RMS criteria for reconstructing the nonparametric estimate of the probability density of a two-dimensional random variable.