The epsilon-skew-normal distribution for analyzing near-normal data

A family of asymmetric distributions, which first appeared in Fechner (1897, Kollectivmasslehre. Leipzig, Engleman) is reparameterized using a skewness parameter ε and named the epsilon-skew-normal family. It is denoted by ESN(θ, σ, ε). Its basic properties such as the relationship between the mean and mode, and its higher-order moments are examined. They are used to obtain simple estimators of the parameters measuring the location θ, the scale σ, and the skewness ε. The maximum likelihood estimates are derived and it is shown that the estimators of θ and σ are asymptotically independent. The estimators reduce properly to the normal case when ε = 0. The ESN(θ, σ, ε) can be used both as a model and as a prior distribution in Bayesian analysis. The posterior distributions in both cases are unimodal, and the modes are available in closed form.