  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
log-likelihood function, both in the univariate and the multivariate context.
