Discriminating between power normal and normal distributions

The Power Normal distribution is a flexible skewed distribution defined on the entire real line, with the Normal distribution arising as a special case. Because of this relationship, the two distributions can often produce very similar probability density and cumulative distribution functions across a wide range of parameter values. While both models may fit the central portion of the data equally well, noticeable differences can appear in the tails, making the choice between them an important statistical problem. This paper investigates the problem of discriminating between the Power Normal and Normal distributions. Unlike most existing discrimination studies, which focus on competing models with the same number of parameters, the present setting involves distributions with different parameter counts, creating additional challenges for model selection. To address this issue, we examine and compare several discrimination criteria, including the ratio of maximized log-likelihoods, a Kolmogorov–Smirnov distance-based approach, and the Bayesian Information Criterion (BIC). The performance of these methods is evaluated through their probabilities of correct selection (PCS). We also derive the asymptotic distributions of selected decision criteria and study the effects of model misspecification. The proposed methods are illustrated using a real data application.