Fuzzy statistical characteristics: modeling and valuing the uncertainty

This paper introduces a novel framework for defining statistical characteristics – such as the mean, quantiles, and expected value – within the context of fuzzy measures and non-additive integration. Motivated by the limitations of classical probability theory when modeling ambiguity, risk distortion, and subjective preferences, we generalize these statistical descriptors using the Choquet integral and functional minimization. Our approach accommodates distorted and fuzzy perceptions of likelihood, aligning naturally with the principles of Prospect Theory, which emphasizes gain–loss asymmetry and cognitive biases in decision-making. By replacing additive measures with fuzzy capacities and employing asymmetric loss functions, we define new forms of fuzzy quantiles and expected values that better reflect real-world uncertainty and individual risk attitudes. Moreover, we demonstrate the applicability of this approach in a portfolio optimization setting under Prospect Theory, where investor behavior is modeled using non-linear utility and probability distortion.