Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

The shifted Chebyshev series-based plug-in for bandwidth selection in kernel density estimation

  • Received : 2023.09.17
  • Accepted : 2024.02.12
  • Published : 2024.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

Kernel density estimation is a prevalent technique employed for nonparametric density estimation, enabling direct estimation from the data itself. This estimation involves two crucial elements: selection of the kernel function and the determination of the appropriate bandwidth. The selection of the bandwidth plays an important role in kernel density estimation, which has been developed over the past decade. A range of methods is available for selecting the bandwidth, including the plug-in bandwidth. In this article, the proposed plug-in bandwidth is introduced, which leverages shifted Chebyshev series-based approximation to determine the optimal bandwidth. Through a simulation study, the performance of the suggested bandwidth is analyzed to reveal its favorable performance across a wide range of distributions and sample sizes compared to alternative bandwidths. The proposed bandwidth is also applied for kernel density estimation on real dataset. The outcomes obtained from the proposed bandwidth indicate a favorable selection. Hence, this article serves as motivation to explore additional plug-in bandwidths that rely on function approximations utilizing alternative series expansions.

Keywords

Acknowledgement

The authors would like to thank Kasetsart University and Rajamangala University of Technology Rattanakosin for the support.

References

  1. Anderson-Cook CM (1999). A tutorial on one-way analysis of circular-linear data, Journal of Quality Technology, 31, 109-119.
  2. Bowman AW (1984). An alternative method of cross-validation for the smoothing of density estimates, Biometrika, 71, 353-360.
  3. Dharmani B (2022). Gram-charlier a series based extended rule-of-thumb for bandwidth selection in univariate kernel density estimation, Austrian Journal of Statistics, 51, 141-163.
  4. Gramacki A (2018). Nonparametric Kernel Density Estimation and Its Computational Aspects, Springer, Switzerland.
  5. Hall P, Sheather SJ, Jones MC, and Marron JS (1991). On optimal data-based bandwidth selection in kernel density estimation, Biometrika, 78, 263-269.
  6. Hardle W (1991). Smoothing Techniques: With Implementation in S, Springer, New York.
  7. Hardle W, Muller M, Sperlich S, and Werwatz A (2004). Nonparametric and Semiparametric Models, Springer, Berlin.
  8. Marron JS and Wand MP (1992). Exact mean integrated squared error, The Annals of Statistics, 20, 712-736.
  9. Park BU and Marron JS (1990). Comparison of data-driven bandwidth selectors, Journal of the American Statistical Association, 85, 66-72.
  10. Parzen E (1962). On estimation of a probability density function and mode, The Annals of Mathematical Statistics, 33, 1065-1076.
  11. Raykar VC and Duraiswami R (2006). Fast optimal bandwidth selection for kernel density estimation, In Proceedings of the 2006 SIAM International Conference on Data Mining, Bethesda, MD, 524-528.
  12. Rosenblatt M (1956). A central limit theorem and a strong mixing condition, Proceedings of the National Academy of Sciences of the United States of America, 42, 43-47.
  13. Rudemo M (1982). Empirical choice of histograms and kernel density estimators, Scandinavian Journal of Statistics, 9, 65-78.
  14. Silverman BW (1986). Density Estimation for Statistics and Data Analysis, Chapman & Hall, London.
  15. Sheather SJ and Jones MC (1991). A reliable data-based bandwidth selection method for kernel density estimation, Journal of the Royal Statistical Society: Series B (Methodological), 53, 683-690.
  16. Tenreiro C (2011). Fourier series-based direct plug-in bandwidth selectors for kernel density estimation, Journal of Nonparametric Statistics, 23, 533-545.
  17. Tenreiro C (2020). Bandwidth selection for kernel density estimation: A Hermite series-based direct plug-in approach, Journal of Statistical Computation and Simulation, 90, 3433-3453.
  18. Wand MP and Jones MC (1995). Kernel Smoothing, Chapman & Hall/CRC, New York.
  19. Woodroofe M (1970). On choosing a delta-sequence, The Annals of Mathematical Statistics, 41, 1665-1671.