Estimation of the integral of the square of derivatives of symmetric probability densities of one-dimensional random variables

When substantiating the method of fast selection of the bandwidth of kernel probability density estimates, a constant was found that is a functional of the second density derivative. In this paper, the obtained result is generalized to derivatives of symmetric probability densities of different orders. The functional dependences of the constants under study on the coefficient of antikurtosis of a random variable are established. The regularities peculiar to them are investigated. Based on the results obtained, a method for estimating functionals from derived probability densities has been developed, which involves the following actions. In the original sample estimated standard deviation of the one-dimensional random variables and the coefficient of antikurtosis. Using the reconstructed functional dependences on the antikurtosis coefficient, the constants are estimated, which are functionals of the derivatives of the probability density. With known estimates of the standard deviation of the investigated random variable and the considered constant, the values of the functional from the derivative of the probability density of the selected order are calculated. The obtained results are confirmed by the analysis of the data of computational experiments. It is established that with increasing order of the derivative, the values of the estimates of the studied functionals increase. This fact is explained by the complication of the integrand function in the considered functionals. The proposed method provides objective results for the first three derivatives of the probability density of a random variable. The obtained conclusions are confirmed by the results of the confidence estimation of the investigated functionals.

функционалы от производных плотности вероятности, симметричные законы распределения, одномерная случайная величина, коэффициент контрэксцесса, functional derivatives of the probability density, symmetric distribution laws, one-dimensional random variable, coefficient of antikurtosis

1. Parzen E., Annals of Mathematical Statistics, 1962, vol. 33, nо. 3, pp. 1065-1076.

2. Епанечников В. А. Непараметрическая оценка многомерной плотности вероятности // Теория вероятности и её применения. 1969. Т. 14. № 1. С. 156-161.

3. Лапко А. В., Лапко В. А. Быстрый алгоритм выбора коэффициентов размытости ядерных функций в непараметрической оценке плотности вероятности // Измерительная техника. 2018. № 6. С. 16-20. DOI: 10.32446/0368-1025it-2018-6-16-20

4. Scott D. W. Multivariate Density Estimation: Theory, Practice, and Visualization. N. J.: John Wiley & Sons, 2015, 384 p.

5. Лапко А. В., Лапко В. А. Зависимость между параметрами гистограммы и ядерной оценки одномодальной плотности вероятности // Измерительная техника. 2019. № 9. С. 3-8. DOI: 10.32446/0368-1025it.2019-9-3-8

6. Лапко А. В., Лапко В. А. Зависимости между параметрами гистограммы и ядерной оценки плотности вероятности многомерной случайной величины // Измерительная техника. 2019. № 11. С. 18-23. DOI: 10.32446/0368-1025it.2019-11-18-23

7. Лапко А. В., Лапко В. А. Быстрый алгоритм выбора коэффициентов размытости в многомерных ядерных оценках плотности вероятности // Измерительная техника. 2018. №10. С. 19-23. DOI: 10.32446/0368-1025it.2018-10-19-23

8. Лапко А. В., Лапко В. А. Методика быстрого выбора коэффициентов размытости ядерных функций в непараметрическом алгоритме распознавания образов // Измерительная техника. 2019. № 4. С. 4-8. DOI: 10.32446/0368-1025it.2019-4-4-8.

9. Лапко А. В., Лапко В. А. Выбор коэффициента размытости ядерных оценок плотности вероятности в условиях больших выборок // Измерительная техника. 2019. № 5. С. 3-6. DOI: 10.32446/0368-1025it.2019-5-3-6

10. Raykar V. C., Duraiswami R., Fast optimal bandwidth selection for kernel density estimation, 6th SIAM International Conference on Data Mining, 2006, pp. 522-526.

11. Rohlfs C., Zahran M., Optimal Bandwidth Selection for Kernel Regression Using a Fast Grid Search and a GPU, 2017 IEEE International parallel and distributed processing symposium workshops (IPDPSW), 2017, pp. 550-556. DOI: 10.1109/IPDPSW.2017.130

12. Silverman B. W., Journal of Royal Statistical society Series C: Applied statistics, 1982, vol. 31, no. 1, pp. 93-99.

13. Sheather S., Jones M., Journal of Royal Statistical Society Series B, 1991, vol. 53, no. 3, pp. 683-690.