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Abstract

Higher-order kernel estimation and kernel density derivative estimation are techniques for reducing the asymptotic mean integrated squared error in nonparametric kernel density estimation. A reduction in the error criterion is an indication of better performance. The estimation of kernel function relies greatly on bandwidth and the identified reduction methods in the literature are bandwidths reliant for their implementation. This study examines the performance of higher order kernel estimation and kernel density derivatives estimation techniques with reference to the Gaussian kernel estimator owing to its wide applicability in real-life-settings. The explicit expressions for the bandwidth selectors of the two techniques in relation to the Gaussian kernel and the bandwidths were accurately obtained. Empirical results using two data sets obviously revealed that kernel density derivative estimation outperformed the higher order kernel estimation excellently well with the asymptotic mean integrated squared error as the criterion function.

Keywords

AMISE Bandwidth Density Derivative Gaussian Estimator Higher Order Kernel

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How to Cite
Israel Uzuazor, S., & Amaju, O. S. (2023). A Comparative Study of Higher Order Kernel Estimation and Kernel Density Derivative Estimation of the Gaussian Kernel Estimator with Data Application. Pakistan Journal of Statistics and Operation Research, 19(2), 299-311. https://doi.org/10.18187/pjsor.v19i2.4233
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References

  1. Azzalini, A. and Bowman, A. W. (1990). A look at some data on the old faithful geyser. Applied Statistics, 39, 357–365.
  2. Bonnier, A., Finne, M. and Weiberg, E. (2019). Examining land-use through GIS-based kernel density estimation: A re-evaluation of legacy data from the Berbati-Limnes Survey. Journal of Field Archaeology, 44(2), 70–83.
  3. Chacón, J. E. and Duong, T. (2013). Data-driven density derivative estimation with applications to nonparametric clustering and bump hunting. Electronic Journal of Statistics, 7,499–532.
  4. Charnigo, R. and Srinivasan, C. (2011). Self-consistent estimation of mean response functions and their derivatives. Canadian Journal of Statistics, 39(2), 280–299
  5. Dobrovidov, A.V. and Rud’ko, I.M. (2010). Bandwidth selection in nonparametric estimator of density derivatives by smoothed cross validation method. Autom. Remote Control, 71, 209–224.
  6. Green, P. J., Latuszyski, K., Pereyra, M. and Robert, C. P. (2015). Bayesian Computation: A summary of the current state, and samples backwards and forwards. Statistics and Computing, 25, 835–862.
  7. Härdle, W., Marron, J. S. and Wand, M. P. (1990). Bandwidth choice for density derivatives. Journal of the Royal Statistical Society, Series B (Statistical Methodology), 52, 223–232.
  8. Helu, A., Samawi, H., Rochani, H., Yin, J. and Vogel, R. (2020). Kernel density estimation based on progressive type-II censoring. Journal of Korean Statistical Society, 49, 475–498.
  9. Henderson, D. J. and Parmeter, C. F. (2015). Applied nonparametric econometrics, Cambridge University Press.
  10. Ishiekwene, C. C. and Osemwenkhae, J. E. (2006). A comparison of forth–order window width selectors in kernel density estimation (A univariate case). ABACUS, 33, 14–20.
  11. Jarnicka, J. (2009). Multivariate kernel density estimation with a parametric support. Opuscula Mathematica, 29(1), 41–45.
  12. Johnson, J. E., Laparra, V., Pérez-Suay, A., Mahecha, M. D. and Camps-Valls, G. (2021). Kernel methods and their derivatives: Concept and perspectives for the Earth System Sciences, https://doi.org/10.1371/journal.pone.0246775.
  13. Jones, M. C. (1992). Potential for automatic bandwidth choice in variations on kernel density estimation. Statistics and Probability Letter, 13, 351–356.
  14. Jones, M. C. and Signorini, D. F. (1997). A comparison of higher–order bias kernel density estimators, Journal of American Statistical Association, 92, 1063–1073.
  15. King, T. L., Bentley, R. J., Thornton, L. E. and Kavanagh, A. M. (2016). Using kernel density estimation to understand the influence of neighborhood destinations on BMI, BMJ, https://doi: 10.1136/bmjopen-2015-008878.
  16. Li, C. L., Chang, W. C., Miroueh, Y., Yang, Y. and Poczos, B. (2019). Implicit kernel learning, Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics (AISTATS), Naha, Okinawa, Japan, PMLR Vol. 89.
  17. Marron, J. S. (1994). Visual understanding of higher-order kernels. Journal of Computational and Graphical Statistics, 3(4), 447–458.
  18. Marron, J. S. and Wand, M. P. (1992). Exact mean integrated squared error. Annals of Statistics, 20, 712–736.
  19. Nan, Y. and Ji, H. (2020). Deep learning for handling kernel/model uncertainty in image deconvolution. Proceedings of the IEEE on Computer Vision and Pattern Recognition (CVPR), pp. 2388–2397.
  20. Parzen, E. (1962). On the estimation of a probability density function and the mode. Annals of Mathematical Statistics, 33, 1065–1076.
  21. Raykar, V. Duraiswami, C. R. and Zhao, L. H. (2015). Fast computation of kernel estimators, Journal of Computational and Graphical Statistics, 19(1), 205–220.
  22. Rojas, F. (2017). An application of kernel estimation method to the demand in two-stage inventory model. Cogent Engineering, https://doi.org/10.1080/23311916.2017.1362803
  23. Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function. Annals of Mathematical Statistics, 27, 832–837.
  24. Sasaki, H., Noh, Y. and Sugiyama, M. (2015). Direct density derivative estimation and its application in KL-Divergence approximation. Proceedings of the 18th International Conference on Artificial Intelligence and Statistics (AISTATS), San Diego, CA, USA. JMLR: W&CP, Volume 38.
  25. Scott, D. W. (2015). Multivariate density estimation: Theory, practice and visualization, Second edition, Wiley, New Jersey.
  26. Siloko, I. U., Ikpotokin, O. and Siloko, E. A. (2019). A key note on performance of smoothing parameterizations in kernel density estimation. Tanzania Journal of Science, 45(1), 1–8
  27. Siloko, I. U., Ishiekwene, C.C. and Oyegue, F.O. (2018). New gradient methods for bandwidth selection in bivariate kernel density estimation. Mathematics and Statistics, 6(1), 1–8.
  28. Siloko, I. U., Ukhurebor, K. E., Siloko, E. A., Enoyoze, E., Bobadoye, A. O., Ishiekwene C. C., Uddin, O. O., and Nwankwo, W. (2021). Effects of some meteorological variables on cassava production in Edo State, Nigeria via density estimation. Scientific African, e00852, https://doi.org/10.1016/j.sciaf.2021.e00852
  29. Siloko, I. U., Nwankwo, W. and Siloko, E. A. (2020a). A new family of kernels from the beta polynomial kernel with applications in density estimation. International Journal of Advances in Intelligent Informatics, 6(3), 235–245.
  30. Siloko, I. U., Siloko, E. A. and Ikpotokin, O. (2020b). A mini review of dimensional effects on asymptotic mean integrated squared error and efficiencies of selected beta kernels. Jordan Journal of Mathematics and Statistics, 13(3), 327–340.
  31. Siloko, I. U. and Siloko, E. A. (2019), Density estimation and data analysis using the kernel method, Proceedings of 56th Annual National Conference (Mathematics Science) of Mathematical Association of Nigeria, 218–226
  32. Silverman, B. W. (2018). Density estimation for statistics and data analysis, Routledge, Abingdon, UK
  33. Somé, S. M. and Kokonendji, C. C. (2021). Bayesian selector of adaptive bandwidth for multivariate gamma kernel estimator on [0,∞)^d, Journal of Applied Statistics, https://doi.org/10.1080/02664763.2021.1881456
  34. Spencer, C. J., Yakymchuk, C. and Ghaznavi, M. (2017). Visualizing data distributions with kernel density estimation and reduced Chi-squared statistics. Geoscience Frontiers, 8(6), 1247–1252.
  35. Stanislaw, W. (2018). Kernel density estimation and its application, ITM Web of Conferences, XLVIII Seminar of Applied Mathematics, 23, 00037.
  36. Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing, Chapman and Hall, London.