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Peer-reviewedCC BY 4.0Mathematics

Exact Credibilistic Expectations of Nonmonotone Fuzzy Transformations with Applications to Higher-Order Moments

  • Hui Li1
  • Xiao Ren2
  • Jian Zhou2,3
  • Yixuan Shen4
  • Zhen Li5
5 affiliations
  1. 1Shengxiang School of Economics and Management, Sanda University, Shanghai 201209, China
  2. 2School of Management, Shanghai University, Shanghai 200444, China
  3. 3Confucius Institute, University of Bahrain, Sakhir P.O. Box 32038, Bahrain
  4. 4School of Business Administration, Hunan University of Technology and Business, Changsha 410205, China
  5. 5School of Logistics and Maritime Studies, Bahrain Polytechnic, Isa Town P.O. Box 33349, Bahrain
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Abstract

Nonmonotone transformations of fuzzy numbers arise in loss functions, dispersion measures, and higher-order risk criteria, but inverse-credibility laws apply directly only to monotone maps. Existing methods typically either compute transformed alpha-cuts without a credibilistic scalar functional or use endpoint formulas whose validity depends on monotonicity. We develop a calculus that preserves interior extrema before integration. For any continuous transformation of a fuzzy number whose transformed output is integrable, the expected value admits an extrema-based alpha-cut representation that requires neither regularity nor an inverse credibility distribution. For continuous nondecreasing radial losses, we derive the exact risk, prove a sharp factor-two bound for the conservative endpoint proxy, characterize both equality regimes, and localize the proxy excess to center-containing cuts. We also establish Lipschitz perturbation and certified midpoint-quadrature bounds. At the decision level, proxy constraints are conservative and proxy minimization has a factor-two guarantee. A reproducible experiment further shows that endpoint substitution can reverse the optimizer and contract the feasible set. The framework is further specialized to higher-order credibilistic moments: for regular fuzzy numbers, signed odd moments retain inverse-distribution formulas, whereas absolute and even moments are obtained from the extrema-based formulation without requiring regularity. Triangular and Gaussian closed forms, standardized coefficients, a scale-free bias function, and deterministic audits demonstrate the resulting error, stability, computation, and decision guarantees.

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Cite

BibTeXRIS
BibTeX
@article{li2026exact,
  title = {{Exact Credibilistic Expectations of Nonmonotone Fuzzy Transformations with Applications to Higher-Order Moments}},
  author = {Hui Li and Xiao Ren and Jian Zhou and Yixuan Shen and Zhen Li},
  journal = {Mathematics},
  year = {2026},
  month = oct,
  doi = {10.3390/math14203768},
  url = {https://www.mdpi.com/2227-7390/14/20/3768},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Exact Credibilistic Expectations of Nonmonotone Fuzzy Transformations with Applications to Higher-Order Moments
AU  - Hui Li
AU  - Xiao Ren
AU  - Jian Zhou
AU  - Yixuan Shen
AU  - Zhen Li
PY  - 2026
DA  - 2026/10/17
JO  - Mathematics
DO  - 10.3390/math14203768
UR  - https://www.mdpi.com/2227-7390/14/20/3768
AB  - Nonmonotone transformations of fuzzy numbers arise in loss functions, dispersion measures, and higher-order risk criteria, but inverse-credibility laws apply directly only to monotone maps. Existing methods typically either compute transformed alpha-cuts without a credibilistic scalar functional or use endpoint formulas whose validity depends on monotonicity. We develop a calculus that preserves interior extrema before integration. For any continuous transformation of a fuzzy number whose transformed output is integrable, the expected value admits an extrema-based alpha-cut representation that requires neither regularity nor an inverse credibility distribution. For continuous nondecreasing radial losses, we derive the exact risk, prove a sharp factor-two bound for the conservative endpoint proxy, characterize both equality regimes, and localize the proxy excess to center-containing cuts. We also establish Lipschitz perturbation and certified midpoint-quadrature bounds. At the decision level, proxy constraints are conservative and proxy minimization has a factor-two guarantee. A reproducible experiment further shows that endpoint substitution can reverse the optimizer and contract the feasible set. The framework is further specialized to higher-order credibilistic moments: for regular fuzzy numbers, signed odd moments retain inverse-distribution formulas, whereas absolute and even moments are obtained from the extrema-based formulation without requiring regularity. Triangular and Gaussian closed forms, standardized coefficients, a scale-free bias function, and deterministic audits demonstrate the resulting error, stability, computation, and decision guarantees.
SN  - 2227-7390
ER  -