Moving average and autoregressive correlation structures under multivariate skew normality

Abstract

This article explores the parameter space of multivariate skew normal families having identical distributed marginal distributions under a few autoregressive-moving average correlation structures (): namely, MA(1), MA(2), and AR(2) correlation structures. Such an undertaking escapes triviality since the efficacious parametrization, in terms of analysis of marginal distributions, restricts the parameter space of the correlation and shape parameters in an intertwined fashion, which upon unraveling illuminates properties of the multivariate skew normal’s correlation matrix, where under identical marginals, Using the parametrization of the multivariate skew normal distribution, the support of δ is found using a limiting argument for the cases considered herein, which can be approximated well via nonlinear regression using a ratio of rational functions of the sample size, k, for an MA(q) correlation structure. Moreover, a range of values for δ is found, when possible, such that the parameter space of the correlation parameters under skew-normality agrees with that under normality. For fixed δ’s, the parameter space of the correlation parameters is found by numerical inversion, where plots have been included to visualize the regions. From the parameter space, inequalities on the component-wise correlations of the skew normal families are developed and displayed pictorially. Finally, transformations are offered to convert the multivariate skew normal families to the more prevalent parametrization, with representing a shape parameter.