A generalized multivariate logistic model and EM algorithm based on the normal variance mean mixture representation

We present an EM algorithm for Maximum Likelihood estimation of the location, scale, and skew, and shape parameters of the z distribution, also known as the generalized logistic function (type IV). We use the Barndorff-Nielsen, Kent, and Sørensen representation of the z distribution as a Gaussian location-scale mixture to derive an EM algorithm for estimating the location, scale, skew, and shape parameters. We use a variational bound on the likelihood function to determine a monotonically converging closed form update for the skew (or drift) parameter. The algorithm also extends naturally to multivariate GLSM estimation using the Kolmogorov-Smirnov mixing density in odd dimensions.