Uni- and multivariate extensions of the sinh-arcsinh normal distribution applied to distributional regression

We first propose to extend the SAS (sinh-arcsinh) normal distribution (Jones and Pewsey, 2009) by allowing the transformed normal random variable to be unstandardized. A Log-SAS transformation for non-negative random variables is then defined, which results in a novel Log-SAS normal distributions. Properties of those distributions are investigated. The SAS transformation can also be applied e.g. to Box-Cox transformed data. Maximum likelihood estimation of the proposed distributions are developed. A chain mixed multivariate extension of the SAS normal distribution and its application to distributional regression are also proposed. Those approaches can e.g. help us to discover possible spurious or hidden bimodal property of a multivariate distribution. The proposals are illustrated by different examples.