Research Article
An Automatic Smoothing Parameter Selection Criterion for Fuzzy Set Density Estimation
Jesús A. Fajardo-González*
Issue:
Volume 15, Issue 5, October 2026
Pages:
79-98
Received:
5 August 2026
Accepted:
19 August 2026
Published:
2 November 2026
DOI:
10.11648/j.pamj.20261505.11
Downloads:
Views:
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.
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 th...
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