The paper studies the problem associated with the optimization of nonparametric probability density estimates, whose relevance is attributed to the lower efficiency of nonparametric algorithms for data processing with the increasing amount of statistical data. In this study, the authors examine a procedure for optimizing the kernel density estimation of a two-dimensional random variable having independent components. The possibility of using the optimal bandwidths of the kernel density estimates of one-dimensional random variables when synthesizing the two-dimensional nonparametric probability density of a random variable having independent components is justified. The proposed approach relies on the asymptotic properties of Rosenblatt–Parzen nonparametric probability density estimation. For a two-dimensional random variable, it is shown that the main contribution to the asymptotic expression for standard deviation is made by the corresponding criteria for one-dimensional random variables. When estimating two-dimensional probability density, it is possible to use bandwidths to minimize the standard deviations of one-dimensional random variables. The obtained conclusions are confirmed by the results of computational experiments in the analysis of normal distribution laws. The possibility of developing the proposed procedure for optimizing the nonparametric probability density estimates of multidimensional random variables having independent components is demonstrated.
Similar content being viewed by others
References
I. V. Zenkov, A. V. Lapko, V. A. Lapko, et al., “A nonparametric algorithm for automatic classification of large multivariate statistical data sets and its application,” Komp. Optika, 45, No. 2, 253–260 (2021), https://doi.org/10.18287/2412-6179-CO-801.
A. G. Varzhapetyan and E. Yu. Mikhailova, “Methods for selecting the key characteristics of nonparametric algorithms for identifying the reliability models of complex systems on the basis of operational data,” Vopr. Kibern., No. 94, 77–87 (1982).
B. W. Silverman, Density Estimation for Statistics and Data Analysis, Chapman and Hall, London (1986).
Z. I. Botev, J. F. Grotowski, and D. P. Kroese, Ann. Stat., 38, No. 5, 2916–2957 (2010).
A. V. Dobrovidov and I. M. Rud’ko, “Selection of the kernel function bandwidth in the nonparametric estimation of derivative density via smoothed cross-validation,” Avtomat. Telemekh., No. 2, 42–58 (2010).
T. A. O’Brien, K. Kashinath, N. R. Cavanaugh, et al., Comp. Stat. Data Anal., 101, 148–160. (2016), https://doi.org/10.1016/j.csda.2016.02.014.
S. Chen, J. Probab. Stat., 2015, 1–21 (2015), https://doi.org/10.1155/2015/242683.
D. W. Scott, Multivariate Density Estimation: Theory, Practice, and Visualization, Wiley, New York (2015).
A. V. Lapko and V. A. Lapko, “Modified algorithm for rapidly determining the bandwidth of kernel density estimation,” Avtometriya, 56, No. 6, 11–18 (2020), https://doi.org/10.15372/AUT20200602.
V. A. Epanechnikov, “Nonparametric estimation of multidimensional probability density,” Teor. Veroyatn. Primen., 14, No. 1, 156–161 (1969).
A. V. Lapko and V. A. Lapko, “Nonparametric probability density estimation of independent random variables,” Inform. Sist. Upravl., 29, No. 3, 118–124 (2011).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Izmeritel’naya Tekhnika, No. 12, pp. 17–21, December, 2021.
Rights and permissions
About this article
Cite this article
Lapko, A.V., Lapko, V.A. & Bakhtina, A.V. Optimization of the Kernel Probability Density Estimation of a Two-Dimensional Random Variable with Independent Components. Meas Tech 64, 958–962 (2022). https://doi.org/10.1007/s11018-022-02029-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11018-022-02029-0