"JOURNAL OF RADIO ELECTRONICS" (Zhurnal Radioelektroniki)  ISSN 1684-1719, N 7, 2018 contents of issue      DOI  10.30898/1684-1719.2018.7.5

Approximation of the empirical probability distributions by Bernstein polynomials

F. V. Golik

Novgorod Branch of the Russian Presidential Academy of National Economy and Public Administration,

31 Germana Street, Veliky Novgorod 173003, Russia

The paper is received on June 28, 2018

Abstract. The present paper is devoted to approximation of the empirical unimodal and multimodal distributions defined on a finite interval. The nonparametric approximation by Bernstein polynomials is studied. A comparative analysis of the optimality criteria is carried out. The criteria minimizing the root-mean-square error of approximation (L2 metric), the uniform metric L, the sigma-metric, the Kullback–Leibler divergence, the Anderson and Darling (AD) criterion, and the sum of the error squares are considered. Instead of AD statistics, which is used in a well-known work by Bradley C. Turnbull, Sujit K. Ghosh (2014), it is suggested to apply the criterion of the least squares method. This allowed to do without solving quadratic programming problems. Optimization of the weight coefficients of the Bernstein polynomial is reduced to solving a linear optimization problem with constraints. Stable and reliable solutions to this problem using the Mathcad software are obtained. Methods for choosing the order of Bernstein polynomial are considered. Search for the optimal order of Bernstein polynomials is carried out according to the generally accepted scheme comprising calculation of an optimal coefficients vector and assessment of approximation error under consistently increasing polynomial order values. The criteria for stopping computations are proposed, under which the order of polynomial is considered optimal. The issue concerning the necessary and sufficient accuracy of the approximation of empirical distributions is discussed. A statistical approach based on interval estimates of the histogram is proposed. The results of computer simulation are presented, which confirm the working ability and high efficiency of the proposed methods of approximation. The results of the work can be applied in solving a wide range of scientific and practical problems related to the analysis of the distribution of empirical data.

Key words: empirical probability distribution; approximation; Bernstein polynomials; least square method; computer simulation.

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For citation:
F.V.Golik. Approximation of the empirical probability distributions by Bernstein polynomials. Zhurnal Radioelektroniki - Journal of Radio Electronics. 2018. No. 7. Available at http://jre.cplire.ru/jre/jul18/5/text.pdf

DOI  10.30898/1684-1719.2018.7.5