$$\phi $$-Extropy in Discrete and Continuous Settings

Abstract The concept of $$\phi $$ ϕ -extropy is introduced as a generalization of extropy for discrete random variables, in analogy with Khinchin’s $$\phi $$ ϕ -entropy. The $$\phi $$ ϕ -extropy framework replaces the logarithmic function in the classical (Shannon) extropy definition with a convex function, yielding a flexible family of uncertainty measures. Its practical utility is demonstrated through applications to classification problems, where it is shown in two examples that performance can improve when using a member of the $$\phi $$ ϕ -family different from the classical extropy. Motivated by a conceptual critique of existing definitions of extropy in the continuous setting, a new definition is proposed based on the maximum of the probability density function and extended to continuous $$\phi $$ ϕ -extropy.