On the cluster of the families of hybrid polynomial kernels in kernel density estimation

Authors

  • Benson Ade Eniola Afere
    Department of Mathematical Sciences, Faculty of Natural Sciences, Prince Abubakar Audu University, 272102, Anyigba, Nigeria

Keywords:

Kernel density estimation, cluster of hybrid kernels, Classical polynomial kernels, global error, Monte Carlo Simulation

Abstract

This study introduces a novel cluster of hybrid polynomial kernel families, designed to achieve significantly lower asymptotic mean integrated squared error compared to traditional kernels. These hybrid kernels are developed by heuristically combining classical polynomial kernels using probability axioms. An in-depth analysis of error propagation within these kernels is conducted, utilizing both simulation experiments and real-life datasets, including the Life Span of Batteries and COVID-19 datasets. The findings consistently demonstrate that the proposed hybrid kernels outperform their classical counterparts in various density estimation tasks across different distribution types and sample sizes. This research highlights the potential of hybrid polynomial kernels to enhance accuracy in density estimation, advocating for their adoption in statistical modelling and analysis.

Dimensions

E. Fix & J. L. Hodges, “Discriminatory analysis: nonparametric discrimination consistency properties”, (Report No. 4, Project No. 21.29.004, USAF School of Aviation Medicine, Randolph Field, Texas, 1951, pp. 238–247. https://sci-hub.se/10.2307/1403797.

H. Akaike, “An approximation to the density functions ”, Annals of the Institute of Statistical Mathematics 6 (1954) 127. https://doi.org/10.1007/BF02900741.

M. Rosenblatt, “Remarks on some nonparametric estimates of a density function ”, Annals of Mathematical Statistic 27 (1956) 832. https://doi.org/10.1007/978-1-4419-8339-8-13.

E. Parzen, “On the estimation of a probability density function and the mode”, Annals of Mathematical Statistics 33 (1962) 1065. https://sci-hub.st/10.1214/aoms/1177704472.

B. W. Silverman, “Density Estimation for Statistics and Data Analysis”, Biometrical Journal 30 (1986) 876. https://doi.org/10.1002/bimj.4710300745.

M. Jiang & S. P. Provost, “A hybrid bandwidth selection methodology for kernel density estimation”, Journal of Statistical Computation and Simulation 84 (2014) 614. https://doi.org/10.1080/00949655.2012.721366.

I. S. Abramson, “On bandwidth variation in kernel estimates–a square root law”, Annals of Mathematical Statistics 27 (1982) 1217. https://doi.org/10.1214/aos/1176345986.

C. J. Stone, “An asymptotically optimal window selection rule for kernel estimates”, Annals of Statistics 12 (1984) 1285. https://doi.org/10.1214/aos/1176346792.

J. S. Marron & W. J. Padget, “Asymptotically optimal bandwidth selection for kernel density estimators from randomly-censored samples”, Annals of Statistics 15 (1987) 1520. https://doi.org/10.1214/aos/1176350607.

M. P. Wand & M. C. Jones, Kernel Smoothing, Chapman & Hall, London, UK, 1994, pp. 1–224. https://doi.org/10.1201/b14876.

J. Simonoff, Smoothing Methods in Statistics, Springer, New York, USA, 1996, pp 1–339. https://link.springer.com/book/10.1007/978-1-4612-4026-6.

A. W. Bowman & A. Azzalini, Applied Smoothing Techniques for Data Analysis: The Kernels Approach with S-Plus Illustration, Oxford University Press, UK, 1997, pp. 1–168. https://doi.org/10.1093/oso/9780198523963.001.0001.

J. E. Chacon, “Data-driven choice of the smoothing parametrization for´ kernel density estimators ”, Canadian Journal of Statistics 37 (2009) 249. https://doi.org/10.1002/cjs.10016.

I. U. Siloko, C. C. Ishiekwene & F. O. Oyegue, ”New gradient methods for bandwidth selection in bivariate kernel density estimation”, Mathematics and Statistics 6 (2018) 1. https://www.hrpub.org/download/20171230/MS1-13409787.pdf.

F. Kimari, A. Adem & L. Kiti, “Efficiency of various bandwidth selection methods across different kernels”, IOSR Journal of Mathematics 15 (2019) 55. https://doi.org/10.9790/5728-1503015562.

B. A. Afere & F. O. Oyegue, “On the construction of the family of ddimensional spherically symmetric polynomial kernels”, Palestine Journal of Mathematics 8 (2019) 286. https://pjm.ppu.edu/sites/default/files/papers/24%20%20%20Adel%20ICMS15b.pdf.

I. U. Siloko, W. Nwankwo & E. A. Siloko, “A new family of kernels from the beta polynomial kernels with applications in density estimation”, International Journal of Advances in Intelligent Informatics 6 (2010) 235. https://www.semanticscholar.org/reader/8d742f2fc7aab2c8ee062f672418231dbd4475f7.

J. Lin & X. Wu, “A hybrid nonparametric multivariate density estimator with applications to risk management”, Econometrics Reviews 42 (2024) 301. https://doi.org/10.1080/07474938.2024.2334119.

B. A. Afere, “A new family of hybrid classical polynomial kernels in density estimation”, International Journal of Research and Innovation Applied Sciences VI (2021) 145. https://rsisinternational.org/journals/ijrias/DigitalLibrary/Vol.6&Issue1/145-151.pdf.

B. A. E. Afere, “On the fourth-order hybrid beta polynomial kernels in kernel density estimation”, Journal of the Nigerian Society of Physical Sciences 6 (2024) 1631. https://doi.org/10.46481/jnsps.2024.1631.

D. W. Scott, Multivariate Density Estimation: Theory, practice and visualization, John Wiley & Sons Inc., New York, USA, 1992, pp. 1–317. https://onlinelibrary.wiley.com/doi/book/10.1002/9780470316849.

J. S. Marron & M. P. Wand, “Exact mean integrated squared error”, Annals of Statistics 20 (1992) 712. https://doi.org/10.1214/aos/1176348653.

A. J. Izenman, Modern Multivariate Statistical Techniques: regression, Classification, and Manifold Learning, Springer, New York, USA, 2008, pp. 1–733. https://link.springer.com/book/10.1007/978-0-387-78189-1.

C. C. Ishiekwene & B. A. E. Afere, “Higher-order window width selectors 15 for emphirical data”, Journal of Nigerian Statistical Association 14 (2001) 69. https://scholar.google.com/citations?user=13DojTQAAAAJ.

P. Osatohanmwen, E. Efe-Eyefia, F. O. Oyegue, J. E. Osemwenkhae, S. M. Ogbonmwan & B. A. Afere, “The exponentiated gumbel–weibull Logistic distribution with application to Nigeria’s COVID–19 infections data”, Annals of Data Science 9 (2022) 909. https://doi.org/10.1007/s40745-022-00373-0.

Published

2025-02-01

How to Cite

On the cluster of the families of hybrid polynomial kernels in kernel density estimation. (2025). Journal of the Nigerian Society of Physical Sciences, 7(1), 2044. https://doi.org/10.46481/jnsps.2025.2044

Issue

Volume 7, Issue 1, February 2025

Section

Mathematics & Statistics
Copyright & Licensing

Copyright (c) 2024 Benson Ade Eniola Afere

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

On the cluster of the families of hybrid polynomial kernels in kernel density estimation. (2025). Journal of the Nigerian Society of Physical Sciences, 7(1), 2044. https://doi.org/10.46481/jnsps.2025.2044

Similar Articles

141-150 of 226

You may also start an advanced similarity search for this article.

Most read articles by the same author(s)