Generalizing the Skew-Normal Distribution: A χν Mixing Approach

This paper formalizes and analyzes a new family of univariate asymmetric distributions generated by a χν mean-mixture of normal variables, extending the classical skew-normal model. The proposed construction introduces a stochastic mean component driven by a chi-distributed latent variable, providing a tractable mechanism for modeling skewness, latent heterogeneity, and additional distributional flexibility. Special cases associated with the Rayleigh (χ2) and Maxwell-Boltzmann (χ3) distributions illustrate the interpretability of the mixing mechanism and its connection with well-known positive distributions. We study the main distributional properties of the proposed family, derive the Fisher information matrix, develop a likelihood-based inference framework, and implement maximum likelihood estimation through an ECME algorithm. The finite-sample performance of the estimators is evaluated by Monte Carlo simulation. Applications to real datasets, including a regression setting, demonstrate that this new class of distributions consistently provides a superior trade-off between complexity and goodness-of-fit compared to traditional asymmetric models.