  The skew-normal and the skew-$t$ distributions are parametric families which
are currently under intense investigation since they provide a more flexible
formulation compared to the classical normal and $t$ distributions by
introducing a parameter which regulates their skewness. While these families
enjoy attractive formal properties from the probability viewpoint, a practical
problem with their usage in applications is the possibility that the maximum
likelihood estimate of the parameter which regulates skewness diverges. This
situation has vanishing probability for increasing sample size, but for finite
samples it occurs with non-negligible probability, and its occurrence has
unpleasant effects on the inferential process. Methods for overcoming this
problem have been put forward both in the classical and in the Bayesian
formulation, but their applicability is restricted to simple situations. We
formulate a proposal based on the idea of penalized likelihood, which has
connections with some of the existing methods, but it applies more generally,
including in the multivariate case.
