On the fourth-order hybrid beta 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, Fourth-order kernels, hybrid kernels, AMISE, Efficiency

Abstract

This paper introduces a novel family of fourth-order hybrid beta polynomial kernels tailored for statistical analysis. The efficacy of these kernels is evaluated using two principal performance metrics: asymptotic mean integrated squared error (AMISE) and kernel efficiency. Comprehensive assessments were conducted using both simulated and real-world datasets, enabling a thorough comparison with conventional fourth-order polynomial kernels. The evaluation process entailed computing AMISE and efficiency metrics for both the hybrid and classical kernels. Consistently, the results illustrated the superior performance of the hybrid kernels over their classical counterparts across diverse datasets, underscoring the robustness and effectiveness of the hybrid approach. By leveraging these performance metrics and conducting evaluations on simulated and real world data, this study furnishes compelling evidence supporting the superiority of the proposed hybrid beta polynomial kernels. The heightened performance, evidenced by lower AMISE values and elevated efficiency scores, strongly advocates for the adoption of the proposed kernels in statistical analysis tasks, presenting a marked improvement over traditional kernels.

Dimensions

W. S. Cleveland & R. McGill, “Graphical perception and graphical methods for analyzing scientific data”, Science 229 (1985) 828. https://doi.org/10.1126/science.229.4716.828.

E. R. Tufte, The Visual display of quantitative information, Graphics Press, Cheshire, 1983. https://doi.org/10.1002/bbpc.19850890519.

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. 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 Statistics 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.

D. W. Scott & S. J. Sheather, “A reliable data-based bandwidth selection method for kernel density estimation”, Journal of the Royal Statistical Society, Series B (Methodological) 53 (1991) 683. https://doi.org/10.1111/j.2517-6161.1991.tb01857.x.

B. W. Silverman, Density Estimation for Statistics and Data Analysis, Chapman & Hall, London, 1986. https://doi.org/10.1002/bimj.4710300745.

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

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

M. P. Wand & M. C. Jones, Kernel Smoothing, Chapman & Hall, London, 1995. https://doi.org/10.1201/b14876.

P. Hall & J. S. Marron, “Lower bounds for bandwidth selection in density estimation”, Probability Theory and Related Fields 90 (1991) 149. https://link.springer.com/article/10.1007/BF01192160.

J. Simonoff, Smoothing methods in statistics, Springer, New York, 1996. https://link.springer.com/book/10.1007/978-1-4612-4026-6.

A. Z. Zambom & R. Dias, “A review of kernel density estimation with applications to econometrics”, arXiv:1212.2812 [stat.ME] 3 (2012) 1. https://doi.org/10.48550/arXiv.1212.2812.

Y. Jeon & J. H. T. Kim, “A gamma kernel density estimation for insurance loss data”, Insurance: Mathematics and Economics 53 (2013) 569. https: //doi.org/10.1016/j.insmatheco.2013.08.009.

X. Wang, M. Wahiduzzaman & A. Yeasmin, “A kernel density estimation approach and statistical generalized additive model of Western North Pacific Typhoon activities”, Atmosphere 13 (2022) 1128. https://doi.org/10.3390/atmos13071128.

J. C. C. Lopez & M. J. Sanz, ”Improving kernels methods for density estimation in random differential equations problems”, Mathematical and Computational Applications, 25 (2020) 33. https://doi.org/10.3390/mca25020033.

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.

V. A. Epanechnikov, “Nonparametric estimation of a multivariate probability density ”, Theory of Probability and its Applications, 14 (1969) 153. https://doi.org/10.1137/1114019.

B. A. Afere, A family of multivariate higher-order hybrid polynomial kernels in kernel density estimation (Doctoral dissertation), University of Benin, Benin City, Nigeria, 2010.

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.

M. G. Kendall & A. Stuart, The Advanced Theory of Statistics, Volume 2, Inference and Relationship, Griffin (1973). https://books.google.com.ng/books/about/The_Advanced_Theory_of_Statistics_Infere.html?id=ogTOAAAAMAAJ&redir_esc=y.

C. C. Ishiekwene & B. A. E. Afere, “Higher-order window width selectors for emphirical data”, Journal of Nigerian Statistical Association 14 (2001) 69.

A. F. Ramsey, “An application of kernel density estimation via diffusion to group yield insurance”, Paper prepared for presentation at the Agricultural & Applied Economics Association’s (AAEA) Annual Meetings, Minneapolis, MN, https://ideas.repec.org/p/ags/aaea14/ 170173.html (2014).

S. J. Shealter, “Density estimation”, Statistical Science 19 (2004) 588. https://doi.org/10.1214/088342304000000297.

B. A. Afere, “A new family of hybrid classical polynomial kernels in density estimation”, International Journal of Research and Innovation in Applied Science (IJRIAS) 4 (2021) 145. https://www.rsisinternational.org/journals/ijrias/DigitalLibrary/Vol.6&Issue1/145-151.pdf.

A. W. Bowman & P. J. Foster, “Adaptive smoothing and density-based tests of multivariate normality ”, Journal of the American Statistical Association 88 (1993) 529. https://doi.org/10.2307/2290333.

W. Hardle, M. Muller, S. Sperlich & A. Werwatz, Nonparametric and Semiparametric models, Springer-Verlag, Berlin, 2004. https://link.springer.com/book/10.1007/978-3-642-17146-8.

M. C. Jones & P. J. Foster, “Generalized Jackknifing and higher-order kernels”, Journal of Nonparametric Statistics 3 (1993) 81. https://doi.org/10.1080/10485259308832573.

B. A. Afere & E. Alih, “On the reduction of global error of multivariate higher-order product polynomial kernels”, Palestine Journal of Mathematics 8 (2019) 286. https://api.semanticscholar.org/CorpusID: 209421718.

B. E. Hansen, Lecture note in nonparametric estimation [Lecture notes], University of Wisconsin, 2004. https://users.ssc.wisc.edu/?bhansen/718/NonParametrics1.pdf.

GeyserTimes, Eruption of old faithful geyser, October 29 (2017) 13:55 [online database], Retrieved from https://geysertimes.org/data.php.

J. Shawe-Taylor & N. Cristianini, Kernel Methods for Pattern Analysis, Cambridge University Press, UK (2004). https://vim.ustc.edu.cn/upload/article/files/d4/72/7d8483bd4f87-ae3f09be1e57664d/a5d0d3ff-de9b-4915-beda-1a2f44357ca2.pdf.

T. Hastie, R. Tibshirani & J. Friedman, The elements of statistical learning: data mining, inference, and prediction, Springer, UK, 2009. https://link.springer.com/book/10.1007/978-0-387-84858-7.

Published

2024-03-06

How to Cite

On the fourth-order hybrid beta polynomial kernels in kernel density estimation. (2024). Journal of the Nigerian Society of Physical Sciences, 6(1), 1631. https://doi.org/10.46481/jnsps.2024.1631

Issue

Volume 6, Issue 1, February 2024

Section

Mathematics & Statistics
Copyright & Licensing

Copyright (c) 2023 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 fourth-order hybrid beta polynomial kernels in kernel density estimation. (2024). Journal of the Nigerian Society of Physical Sciences, 6(1), 1631. https://doi.org/10.46481/jnsps.2024.1631

Similar Articles

181-190 of 203

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

Most read articles by the same author(s)