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Analysis of the ratio of the mean square deviations of the kernel probability density estimation in the conditions of independent and dependent random variables
https://doi.org/10.32446/0368-1025it.2021-3-9-14
Abstract
The influence on the approximation properties of a nonparametric probability density estimate of Rosenblatt-Parzen type of the information on the dependence of random variables is determined. The ratio of the asymptotic expressions of the mean square deviations of independent and dependent random variables is obtained. This relation for a two-dimensional random variable is considered as a quantitative assessment of the influence of information about their dependence on the approximation properties of the kernel probability density estimate. The established ratio is determined by the kind of probability density and the volumes of the initial statistical data that are used in estimating the probability densities of dependent and independent random variables. The general results obtained are considered in detail for two-dimensional linearly dependent random variables with normal distribution laws. The functional dependence of the ratio of the mean square deviations of the independent and dependent two-dimensional random variables on the correlation coefficient is determined. The dependence of the considered ratio on the volume of statistical data is analyzed. A method for estimating the functional of the second derivatives of two-dimensional random variables with normal distribution laws is developed. The results obtained are the basis for the development of modifi cations of “fast” procedures for optimizing kernel estimates of probability densities in conditions of large samples.
About the Authors
A. V. Lapko
Institute of Computational Modeling, Siberian Branch of the Russian Academy of Sciences (ICM SB RAS);
Reshetnev Siberian State University of Science and Technology
Russian Federation
Russian Federation
Aleksandr V. Lapko
Krasnoyarsk
V. A. Lapko
Institute of Computational Modeling, Siberian Branch of the Russian Academy of Sciences (ICM SB RAS);
Reshetnev Siberian State University of Science and Technology
Russian Federation
Russian Federation
Vasiliy A. Lapko
Krasnoyarsk
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Review
For citations:
Lapko A.V., Lapko V.A. Analysis of the ratio of the mean square deviations of the kernel probability density estimation in the conditions of independent and dependent random variables. Izmeritel`naya Tekhnika. 2021;(3):9-14. (In Russ.) https://doi.org/10.32446/0368-1025it.2021-3-9-14
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