Quantile-Based Skewness for Fuzzy Numbers with Probabilistic Foundations: With an Application in Portfolio Optimization

Abstract

This paper introduces a novel parameter free skewness coefficient for fuzzy numbers, addressing a critical gap in quantifying asymmetry under imprecision. Existing fuzzy literature substitutes membership functions for probability density functions in moment-based skewness, lacking rigorous theoretical grounding. Our coefficient, however, rigorously establishes a probabilistic foundation, making it both probabilistically meaningful and fully compliant with the semantics of fuzzy set theory. Our approach interprets a fuzzy number's left and right membership function components as cumulative and survival probability functions of associated random variables. This provides a robust probabilistic foundation for its $α$-cuts as generalized quantiles representing values that are "at least $α$-probable", thereby instantiating the well-grounded dualism between probability and possibility theory. As a quantile-based measure, the proposed coefficient offers invariance under scale and location transformations. Crucially, its superior computational efficiency, empirically demonstrating an almost logarithmic reduction in portfolio optimization processing time with increasing assets, enables significant scalability for real-world applications. The coefficient comprises two complementary constituents: an "inner" measure quantifying the intrinsic skewness of the underlying probabilistic distributions, and an "outer" measure capturing the fuzzy number's overall profile asymmetry. This dual structure offers nuanced insights into a fuzzy number's asymmetry. We demonstrate its practical utility and computational advantages within a fuzzy mean-variance-skewness portfolio optimization framework, comparing its performance with two of the most highly cited original moment-based fuzzy skewness coefficients from the literature.

Figures (4)

• Figure 1: Example \ref{['ex:FN_to_RV']}: Left panel: piecewise fuzzy number $\xi$. Middle panel: the left component $\xi_l$ of the fuzzy number $\xi$, interpreted as the CDF of a random variable $X_L$. The right component $\xi_r$ of the fuzzy number $\xi$, interpreted as the survival function $1 - \text{CDF} = \Pr\{X_R > x\}$ of a random variable $X_R$. These interpretations align with the associations established in \ref{['eq:leftComponent2leftRandom']} and \ref{['eq:rightComponent2rightRandom']}. Right panel: the associated probability density functions (PDFs) of $X_L$ and $X_R$, respectively, derived from the membership function components: $f_{X_L}(x) = \frac{d\xi_l(x)}{dx}$ and $f_{X_R}(x) = \frac{d(1-\xi_r(x))}{dx}$. The numerical values for the discussed skewness coefficients of $\xi$ are provided in Example \ref{['ex:FN_to_RV']} in the main text.
• Figure 2: Visualization for Example \ref{['ex:RV_to_FN_Visualization']}. (Top) PDFs of the generative Beta distributions. (Middle) CDFs forming the left and right slopes. (Bottom) The resulting membership function $\xi(x)$. The x-axis annotations highlight the core boundaries and the specific quantile points used for the point-based skewness calculation.
• Figure 3: Schematic of Threshold-Constrained Portfolio Optimization for a portfolio with three assets modeled by fuzzy numbers $\xi_1$, $\xi_2$, and $\xi_3$. In the first step expected value, variance, and skewness are computed for each weight triplet (not all possible weight triplets are shown here) of the mesh. In the second step two of the three parameters are fixed according to the decision maker's preference. For the reduced subset, the weight triplet that maximizes skewness is selected.
• Figure 4: The dependency of portfolio optimization computational time (in seconds), where panels (a), (b), and (c) correspond to the three different versions of the model \ref{['eq:PortfolioOptimization']}, see Remark \ref{['rem:DifferentObjectives']}: (a) minimizing variance, (b) maximizing skewness, or (c) maximizing expected value, respectively. Colors indicate different definitions of skewness: $\mathop{\mathrm{\mathbb{S}}}\nolimits_{LGY15}$ (li2015skewness) is blue, $\mathop{\mathrm{\mathbb{S}}}\nolimits_{VB13}$ (VB13) is yellow, and our $\mathop{\mathrm{\mathbb{S}}}\nolimits_{JKPT_1(0.5)(0.25)}$ is red.

Theorems & Definitions (9)

• Definition 1
• Remark 2.1
• Definition 2
• Definition 3
• Example 3.1
• Example 3.2
• Example 3.3
• Remark 4.1
• Remark 4.2