Minimum Hellinger distance based inference for scalar skew-normal and skew-t distributions

Abstract

The skew-normal is a parametric model that extends the normal family by the addition of a shape parameter to account for skewness. As well, the skew-t distribution is generated by a perturbation of symmetry of the basic Student’s t density. These families share some nice properties. In particular, they allow a continuous variation through different degrees of asymmetry and, in the case of the skew-t, tail thickness, but still retain relevant features of the perturbed symmetric densities. In both models, a problem occurs in the estimation of the skewness parameter: for small and moderate sample sizes, the maximum likelihood method gives rise to an infinite estimate with positive probability, even when the sample skewness is not too large. To get around this phenomenon, we consider the minimum Hellinger distance estimation technique as an alternative to maximum likelihood. The method always leads to a finite estimate of the shape parameter. Furthermore, the procedure is asymptotically efficient under the assumed model and allows for testing hypothesis and setting confidence regions in a standard fashion.