ESTIMATION OF A NONLINEAR FUNCTIONAL OF THE PROBABILITY DENSITY OF A THREE-DIMENSIONAL RANDOM VARIABLE FOR PROBLEMS OF FAST OPTIMIZATION OF NONPARAMETRIC STATISTICS
A. V. Lapko1,2, V. A. Lapko1,2
1Institute of Computational Modelling, Siberian Branch, Russian Academy of Sciences, Krasnoyarsk, Russia
2Reshetnev Siberian State University of Science and Technology, Krasnoyarsk, Russia
Keywords: kernel probability density estimation, nonlinear probability density functional, three-dimensional random variable, fast bandwidths selection, antikurtosis coefficient, skewness coefficient, large sampling, lognormal distribution law
Abstract
A method for estimating the nonlinear functional of the probability density of a three-dimensional random variable is proposed. It is relevant in the implementation of procedures for fast bandwidths selection in the problem of optimizing kernel estimates of the probability density. Solving this problem can significantly improve the computational efficiency of nonparametric decision rules. The proposed approach is based on the analysis of the formula for the optimal bandwidths of the kernel probability density estimate. The bandwidths of the kernel functions are represented as the product of an undefined parameter and the standard deviations of the analyzed random variables. The main component of the undefined parameter is a nonlinear functional of the probability density. The considered functional for a family of unimodal distribution laws is determined by the form of the probability density and does not depend on the density parameters. It is determined by the approximation of the functional dependence on the antikurtosis and skewness coefficients, which are estimated from the initial statistical data. To simplify the problem of restoring the desired dependence, the antikurtosis and skewness coefficients are transformed into a generalized parameter. The initial information is made up of a family of lognormal distribution laws. The errors of approximation of the considered nonlinear functional of the probability density are estimated by the values of the introduced generalized parameter for a family of three-dimensional lognormal distribution laws of independent random variables. The possibility of using the proposed methodology for estimating nonlinear functionals of the probability densities that differ from lognormal distribution laws is investigated. The influence of the arising approximation errors on the mean square criteria for recovering a nonparametric probability density estimate of a three-dimensional random variable is analyzed.
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