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The skew-normal and the skew-$t$ distributions are parametric families whichare currently under intense investigation since they provide a more flexibleformulation compared to the classical normal and $t$ distributions byintroducing a parameter which regulates their skewness. While these familiesenjoy attractive formal properties from the probability viewpoint, a practicalproblem with their usage in applications is the possibility that the maximumlikelihood estimate of the parameter which regulates skewness diverges. Thissituation has vanishing probability for increasing sample size, but for finitesamples it occurs with non-negligible probability, and its occurrence hasunpleasant effects on the inferential process. Methods for overcoming thisproblem have been put forward both in the classical and in the Bayesianformulation, but their applicability is restricted to simple situations. Weformulate a proposal based on the idea of penalized likelihood, which hasconnections with some of the existing methods, but it applies more generally,including in the multivariate case.

Maximum penalized likelihood estimation for skew-normal and skew-$t$ distributions