Comparison of the Effectiveness of Methods for Sampling the Range of Variation of Random Quantities in Synthesis of Nonparametric Estimates of Probability Density

Abstract

The approximation properties of a nonparametric estimate of probability density are studied for different methods of sampling the domain of definition. The indicators of the effectiveness of these methods are estimated. Keywords: nonparametric estimate of probability density, approximation properties, sampling methods, normal distribution law.The computational efficiency of nonparametric algorithms for data processing are determined by the amount of statistical data and decrease with increasing amounts of data. Under these conditions, it is appropriate to use both the principles of decomposition of the initial statistical data based on their volume and parallel computing. A mixture of nonparametric estimates of the probability densities for univariate and multivariate random quantities has been proposed and studied [1, 2]. It has a considerably smaller dispersion than the traditional Rosenblatt-Parzen estimate of the probability density [3]. The reduction in computational time is comparable to the number of components of the mixture of nonparametric probability density estimates.These results have been generalized for estimating the resolving function in the problem of pattern recognition under large sample conditions. Two-level nonparametric systems have been developed for solving two-and multialternative classification problems, and the asymptotic properties of estimates of the equations of separating surfaces for the univariate and multivariate cases have been established [4][5][6][7][8]. The problem of large samples can be avoided by using a nonparametric estimate of the probability density, the synthesis of which is based on compressing the initial statistical data [9][10][11]. The condition of a minimum asymptotic expression for its mean square deviation (MSD) is used to determine a procedure for optimal choice of the number of sampling intervals in the range of variation of the random variable [12].Here an analysis of the approximation properties of a nonparametric estimate of the probability density is used to compare the most widespread procedures for sampling the range of variation of normally distributed random quantities.Synthesis of a Nonparametric Estimate of the Probability Density. Consider a sample V = (x i , i =⎯⎯⎯ 1, n) of n independent values of a univariate random quantity x with an unknown probability density p(x). We divide the region over which p(x) is to be determined into N nonintersecting intervals of length 2β and form a set of random quantities X j , j =⎯⎯⎯ 1, n. As the

Modification of Nonparametric Probability Density Estimation of Parzen's Type with Implicitly Given form of Kernel Function

“…Nonparametric probability density estimates which are based on the procedure of compression of input statistical data are developed for conditions of selections of large volume [4]. The methods of optimum sampling of a range of values of random values [5] and confidence estimation of their probability density are offered on this basis [6].…”