Skip to main content

Optimization of the Kernel Probability Density Estimation of a Two-Dimensional Random Variable with Independent Components

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.

This is a preview of subscription content, access via your institution.

References

  1. 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.

    ADS  Article  Google Scholar 

  2. 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).

  3. B. W. Silverman, Density Estimation for Statistics and Data Analysis, Chapman and Hall, London (1986).

    MATH  Google Scholar 

  4. Z. I. Botev, J. F. Grotowski, and D. P. Kroese, Ann. Stat., 38, No. 5, 2916–2957 (2010).

    Article  Google Scholar 

  5. 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).

  6. 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.

    Article  Google Scholar 

  7. S. Chen, J. Probab. Stat., 2015, 1–21 (2015), https://doi.org/10.1155/2015/242683.

    Article  Google Scholar 

  8. D. W. Scott, Multivariate Density Estimation: Theory, Practice, and Visualization, Wiley, New York (2015).

    MATH  Google Scholar 

  9. 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.

    Article  Google Scholar 

  10. V. A. Epanechnikov, “Nonparametric estimation of multidimensional probability density,” Teor. Veroyatn. Primen., 14, No. 1, 156–161 (1969).

    MathSciNet  MATH  Google Scholar 

  11. A. V. Lapko and V. A. Lapko, “Nonparametric probability density estimation of independent random variables,” Inform. Sist. Upravl., 29, No. 3, 118–124 (2011).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. V. Lapko.

Additional information

Translated from Izmeritel’naya Tekhnika, No. 12, pp. 17–21, December, 2021.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11018-022-02029-0

Keywords

  • independent random variables
  • probability density
  • nonparametric estimation
  • optimization of estimates
  • kernel functions
  • bandwidth
  • asymptotic properties