q-Symmetric Family of Continuous Distributions‎: Construction‎, ‎Properties‎, ‎and Measures of Skewness‎

Accepted February 2026

Abstract

‎We introduce in this work a new family of probability distributions that generalizes the classical classes of location-symmetric and log-symmetric distributions‎. ‎This family‎, ‎termed q-symmetric distributions‎, ‎is defined through a monotone involution transformation‎, ‎referred to as the q-transformation‎. ‎We then present some properties of the q-transformation and investigate systematic methods for constructing q-symmetric distributions‎. ‎Several theoretical properties of the proposed family are established and‎, ‎in addition‎, ‎we introduce some measures of skewness associated with the q-transformation‎, ‎including both moment-based and quantile-based measures‎, ‎extending classical concepts of skewness to the q-symmetric setting considered here‎. ‎The main properties of these q-skewness measures are then studied and illustrated‎.