A multivariate skew-normal-Tukey-h distribution

We introduce a new family of multivariate distributions by taking the component-wise Tukey-$h$ transformation of a random vector following a skew-normal distribution with an alternative parameterization. The proposed distribution is named the skew-normal-Tukey-$h$ distribution and is an extension of the skew-normal distribution for handling heavy-tailed data. We compare this proposed distribution to the skew-$t$ distribution, which is another extension of the skew-normal distribution for modeling tail-thickness, and demonstrate that when there are substantial differences in marginal kurtosis, the proposed distribution is more appropriate. Moreover, we derive many appealing stochastic properties of the proposed distribution and provide a methodology for the estimation of the parameters that can be applied to large dimensions. Using simulations, as well as a wine and a wind speed data application, we illustrate how to draw inferences based on the multivariate skew-normal-Tukey-$h$ distribution.