Some computational aspects of maximum likelihood estimation of the skew-$t$ distribution

Abstract

Since its introduction, the skew-t distribution has received much attention in the literature both for the study of theoretical properties and as a model for data fitting in empirical work. A major motivation for this interest is the high degree of flexibility of the distribution as the parameters span their admissible range, with ample variation of the associated measures of skewness and kurtosis. While this high flexibility allows to adapt a member of the parametric family to a wide range of data patterns, it also implies that parameter estimation is a more delicate operation with respect to less flexible parametric families, given that a small variation of the parameters can have a substantial effect on the selected distribution. In this context, the aim of the present contribution is to deal with some computational aspects of maximum likelihood estimation. A problem of interest is the possible presence of multiple local maxima of the log-likelihood function. Another one, to which most of our attention is dedicated, is the development of a quick and reliable initialization method for the subsequent numerical maximization of the loglikelihood function, both in the univariate and the multivariate context.