Mathematics & Physics 2017, 10(1), 16–21 УДК 519.24 Improving the Accuracy of the Probability Density Function Estimation Boris S. Dobronets∗ Olga A.Popova† Institute of Space and Information Technology Siberian Federal University Kirenskogo, 26, Krasnoyarsk, 660074 Russia Received 03.06.2016, received in revised form 09.09.2016, accepted 10.11.2016 The paper considers the new approach to the reconstruction of the probability density function similarly the averaged shifted histogram method. <...> An algorithm is used Richardson’s extrapolation for increasing accuracy. <...> We prove the estimates of the accuracy of the probability density function and its second derivative to choose the optimal settings for smoothing the histogram and kernel estimators and to consider the optimal choice problem of the bandwidth parameter. <...> Keywords: MISE, error estimate, Richardson’s extrapolation, Runge’s rule, probability density functions estimation, probability density function derivatives, Numerical probabilistic analysis. <...> Important to know the estimate for mathematical expectation of norm error of the constructed empirical probability density function [4]. <...> This paper considers the Runge’s rule application to the calculation the second derivative estimates of the probability density function. <...> In contrast to the known methods, this approach does not require the differentiation of kernel estimates or calculations of finite differences from empirical probability density function. <...> Use of estimates of the second derivatives allows to obtain realistic estimates of the mathematical expectations of l2 error norm for the probability density function reconstruction. <...> Knowledge of these assessments allows us to calculate the optimal bandwidth parameter h [5]. <...> One of the first rules for practical evaluation of the error, was proposed by K.Runge in the beginning of the XX century. <...> The integer k characterizes the order ∗BDobronets@yandex.ru †OlgaArc@yandex.ru ⃝ Siberian Federal University. <...> All rights reserved c – 16 – Boris S. Dobronets, Olga A. Popova Improving the Accuracy of the Probability Density Function . . . of accuracy of the approximate solution, and m > 0 gives smallness of the remainder term as compared to the major error term hkv. <...> Hence, the major error term can be determined: uh/2 −u ≈ uh −uh/2 2k −1 . (3) Since in formula (3), the remainder term of order O(hk+m) is rejected, it <...>