An Automatic Smoothing Parameter Selection Criterion for Fuzzy Set Density Estimation

Jesús A. Fajardo-González

doi:doi:10.11648/j.pamj.20261505.11

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An Automatic Smoothing Parameter Selection Criterion for Fuzzy Set Density Estimation

Jesús A. Fajardo-González^*

Published in Pure and Applied Mathematics Journal (Volume 15, Issue 5)

Received: 5 August 2026 Accepted: 19 August 2026 Published: 2 November 2026

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Abstract

The fuzzy set density estimator of a probability density function is an innovative non-kernel-based estimator that exhibits superior performance compared to the classical kernel density estimator. The numerical values assigned to the smoothing parameters directly govern the performance and precision of both estimators; however, although explicit theoretical formulas exist to compute their optimal global smoothing parameters, these expressions inherently depend on knowledge of the underlying density function. This fundamental constraint renders bandwidth selection the primary operational challenge when estimating an unknown probability density function using the fuzzy set density estimator, in stark contrast to the kernel density estimator, which benefits from a wide variety of well-established parameter selection criteria in statistical literature. To address this structural limitation, this paper introduces the risk function estimator corresponding to the integrated mean squared error within the general framework of fuzzy set density estimation. Consequently, the proposed estimator is fully equipped with an automatic and autonomous smoothing parameter selection criterion, which remains fully applicable regardless of whether the density function is known or completely unknown. The theoretical findings significantly expand the practical applicability of the fuzzy set density estimator and represent a milestone contribution to fuzzy density estimation methodologies by providing its first automatic data-driven bandwidth selection rule. Finally, the theoretical properties are thoroughly illustrated through a detailed numerical example and two real-data applications.

Published in	Pure and Applied Mathematics Journal (Volume 15, Issue 5)
DOI	10.11648/j.pamj.20261505.11
Page(s)	79-98
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Fuzzy Set Estimator, Non-parametric Density Estimation, Smoothing Parameter Selection, Risk Function Estimation, Cross-validation

References

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[3]	J. Fajardo and P. Harmath, "Boundary estimation with the fuzzy set density estimator," METRON, 79(3): 285–302, 2021. https://doi.org/10.1007/s40300-021-00210-z
[4]	J. Fajardo, R. Ríos, and L. Rodríguez, "Properties of convergence of a fuzzy set estimator of the density function," Braz. J. Probab. Stat., 26(2): 208–217, 2012. http://www.jstor.org/stable/43601209
[5]	J. A. Fajardo, "Notas sobre el desempeño de los estimadores fronteras de la densidad con núcleo localmente adaptable y con conjunto difuso," Divulg. Mat., 22(2): 48–65, 2022. https://doi.org/10.5281/zenodo.7487484
[6]	M. Falk and F. Liese, "LAN of thinned empirical processes with an application to fuzzy set density estimation," Extremes, 1(3): 323–349, 1998. https://doi.org/10.1023/A:1009981817526
[7]	A. Gramacki, Nonparametric Kernel Density Estimation and Its Computational Aspects, volume 37. Springer, 2018. https://doi.org/10.1007/978-3-319-71688-6
[8]	N.-B. Heidenreich, A. Schindler, and S. Sperlich, "Bandwidth selection for kernel density estimation: a review of fully automatic selectors," AStA Adv. Stat. Anal., 97(4): 403–433, 2013. https://doi.org/10.1007/s10182-013-0216-y
[9]	W. Hoeffding, "A class of statistics with asymptotically normal distribution," Ann. Math. Statist., 19(3): 293–325, 1948. http://www.jstor.org/stable/2235637
[10]	R. G. Jarrett, "A note on the intervals between coalmining disasters," Biometrika, 66(1): 191–193, 1979. https://doi.org/10.1093/biomet/66.1.191
[11]	E. Parzen, "On estimation of a probability density function and mode," Ann. Math. Statist., 33(3): 1065–1076, 1962. http://www.jstor.org/stable/2237880
[12]	R. D. Reiss, A Course on Point Processes. Springer Series in Statistics, Springer, New York, 1993. https://doi.org/10.1007/978-1-4613-9308-5
[13]	M. Rosenblatt, "Remarks on some nonparametric estimates of a density function," Ann. Math. Statist., 27(3): 832–837, 1956. https://doi.org/10.1214/aoms/1177728190
[14]	M. Rudemo, "Empirical choice of histograms and kernel density estimators," Scand. J. Statist., 9(2): 65–78, 1982. https://www.jstor.org/stable/4615859
[15]	B. W. Silverman, Density Estimation for Statistics and Data Analysis. Chapman and Hall, New York, 1986. https://doi.org/10.1007/978-1-4899-3324-9
[16]	M. P. Wand and M. C. Jones, Kernel Smoothing (1st ed.). Chapman and Hall/CRC, 1994. https://doi.org/10.1201/b14876

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APA Style

Fajardo-González, J. A. (2026). An Automatic Smoothing Parameter Selection Criterion for Fuzzy Set Density Estimation. Pure and Applied Mathematics Journal, 15(5), 79-98. https://doi.org/10.11648/j.pamj.20261505.11

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ACS Style

Fajardo-González, J. A. An Automatic Smoothing Parameter Selection Criterion for Fuzzy Set Density Estimation. Pure Appl. Math. J. 2026, 15(5), 79-98. doi: 10.11648/j.pamj.20261505.11

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AMA Style

Fajardo-González JA. An Automatic Smoothing Parameter Selection Criterion for Fuzzy Set Density Estimation. Pure Appl Math J. 2026;15(5):79-98. doi: 10.11648/j.pamj.20261505.11

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@article{10.11648/j.pamj.20261505.11,
  author = {Jesús A. Fajardo-González},
  title = {An Automatic Smoothing Parameter Selection Criterion for Fuzzy Set Density Estimation},
  journal = {Pure and Applied Mathematics Journal},
  volume = {15},
  number = {5},
  pages = {79-98},
  doi = {10.11648/j.pamj.20261505.11},
  url = {https://doi.org/10.11648/j.pamj.20261505.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20261505.11},
  abstract = {The fuzzy set density estimator of a probability density function is an innovative non-kernel-based estimator that exhibits superior performance compared to the classical kernel density estimator. The numerical values assigned to the smoothing parameters directly govern the performance and precision of both estimators; however, although explicit theoretical formulas exist to compute their optimal global smoothing parameters, these expressions inherently depend on knowledge of the underlying density function. This fundamental constraint renders bandwidth selection the primary operational challenge when estimating an unknown probability density function using the fuzzy set density estimator, in stark contrast to the kernel density estimator, which benefits from a wide variety of well-established parameter selection criteria in statistical literature. To address this structural limitation, this paper introduces the risk function estimator corresponding to the integrated mean squared error within the general framework of fuzzy set density estimation. Consequently, the proposed estimator is fully equipped with an automatic and autonomous smoothing parameter selection criterion, which remains fully applicable regardless of whether the density function is known or completely unknown. The theoretical findings significantly expand the practical applicability of the fuzzy set density estimator and represent a milestone contribution to fuzzy density estimation methodologies by providing its first automatic data-driven bandwidth selection rule. Finally, the theoretical properties are thoroughly illustrated through a detailed numerical example and two real-data applications.},
 year = {2026}
}

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TY  - JOUR
T1  - An Automatic Smoothing Parameter Selection Criterion for Fuzzy Set Density Estimation
AU  - Jesús A. Fajardo-González
Y1  - 2026/11/02
PY  - 2026
N1  - https://doi.org/10.11648/j.pamj.20261505.11
DO  - 10.11648/j.pamj.20261505.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 79
EP  - 98
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20261505.11
AB  - The fuzzy set density estimator of a probability density function is an innovative non-kernel-based estimator that exhibits superior performance compared to the classical kernel density estimator. The numerical values assigned to the smoothing parameters directly govern the performance and precision of both estimators; however, although explicit theoretical formulas exist to compute their optimal global smoothing parameters, these expressions inherently depend on knowledge of the underlying density function. This fundamental constraint renders bandwidth selection the primary operational challenge when estimating an unknown probability density function using the fuzzy set density estimator, in stark contrast to the kernel density estimator, which benefits from a wide variety of well-established parameter selection criteria in statistical literature. To address this structural limitation, this paper introduces the risk function estimator corresponding to the integrated mean squared error within the general framework of fuzzy set density estimation. Consequently, the proposed estimator is fully equipped with an automatic and autonomous smoothing parameter selection criterion, which remains fully applicable regardless of whether the density function is known or completely unknown. The theoretical findings significantly expand the practical applicability of the fuzzy set density estimator and represent a milestone contribution to fuzzy density estimation methodologies by providing its first automatic data-driven bandwidth selection rule. Finally, the theoretical properties are thoroughly illustrated through a detailed numerical example and two real-data applications.
VL  - 15
IS  - 5
ER  -

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Author Information

Jesús A. Fajardo-González

Department of Mathematics, University of Oriente, Cumana, Venezuela

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http://orcid.org/0000-0002-3762-4824

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APA Style

Fajardo-González, J. A. (2026). An Automatic Smoothing Parameter Selection Criterion for Fuzzy Set Density Estimation. Pure and Applied Mathematics Journal, 15(5), 79-98. https://doi.org/10.11648/j.pamj.20261505.11

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ACS Style

Fajardo-González, J. A. An Automatic Smoothing Parameter Selection Criterion for Fuzzy Set Density Estimation. Pure Appl. Math. J. 2026, 15(5), 79-98. doi: 10.11648/j.pamj.20261505.11

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AMA Style

Fajardo-González JA. An Automatic Smoothing Parameter Selection Criterion for Fuzzy Set Density Estimation. Pure Appl Math J. 2026;15(5):79-98. doi: 10.11648/j.pamj.20261505.11

Copy | Download

@article{10.11648/j.pamj.20261505.11,
  author = {Jesús A. Fajardo-González},
  title = {An Automatic Smoothing Parameter Selection Criterion for Fuzzy Set Density Estimation},
  journal = {Pure and Applied Mathematics Journal},
  volume = {15},
  number = {5},
  pages = {79-98},
  doi = {10.11648/j.pamj.20261505.11},
  url = {https://doi.org/10.11648/j.pamj.20261505.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20261505.11},
  abstract = {The fuzzy set density estimator of a probability density function is an innovative non-kernel-based estimator that exhibits superior performance compared to the classical kernel density estimator. The numerical values assigned to the smoothing parameters directly govern the performance and precision of both estimators; however, although explicit theoretical formulas exist to compute their optimal global smoothing parameters, these expressions inherently depend on knowledge of the underlying density function. This fundamental constraint renders bandwidth selection the primary operational challenge when estimating an unknown probability density function using the fuzzy set density estimator, in stark contrast to the kernel density estimator, which benefits from a wide variety of well-established parameter selection criteria in statistical literature. To address this structural limitation, this paper introduces the risk function estimator corresponding to the integrated mean squared error within the general framework of fuzzy set density estimation. Consequently, the proposed estimator is fully equipped with an automatic and autonomous smoothing parameter selection criterion, which remains fully applicable regardless of whether the density function is known or completely unknown. The theoretical findings significantly expand the practical applicability of the fuzzy set density estimator and represent a milestone contribution to fuzzy density estimation methodologies by providing its first automatic data-driven bandwidth selection rule. Finally, the theoretical properties are thoroughly illustrated through a detailed numerical example and two real-data applications.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - An Automatic Smoothing Parameter Selection Criterion for Fuzzy Set Density Estimation
AU  - Jesús A. Fajardo-González
Y1  - 2026/11/02
PY  - 2026
N1  - https://doi.org/10.11648/j.pamj.20261505.11
DO  - 10.11648/j.pamj.20261505.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 79
EP  - 98
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20261505.11
AB  - The fuzzy set density estimator of a probability density function is an innovative non-kernel-based estimator that exhibits superior performance compared to the classical kernel density estimator. The numerical values assigned to the smoothing parameters directly govern the performance and precision of both estimators; however, although explicit theoretical formulas exist to compute their optimal global smoothing parameters, these expressions inherently depend on knowledge of the underlying density function. This fundamental constraint renders bandwidth selection the primary operational challenge when estimating an unknown probability density function using the fuzzy set density estimator, in stark contrast to the kernel density estimator, which benefits from a wide variety of well-established parameter selection criteria in statistical literature. To address this structural limitation, this paper introduces the risk function estimator corresponding to the integrated mean squared error within the general framework of fuzzy set density estimation. Consequently, the proposed estimator is fully equipped with an automatic and autonomous smoothing parameter selection criterion, which remains fully applicable regardless of whether the density function is known or completely unknown. The theoretical findings significantly expand the practical applicability of the fuzzy set density estimator and represent a milestone contribution to fuzzy density estimation methodologies by providing its first automatic data-driven bandwidth selection rule. Finally, the theoretical properties are thoroughly illustrated through a detailed numerical example and two real-data applications.
VL  - 15
IS  - 5
ER  -

Copy | Download