Testing for sub-models of the skew t-distribution

The skew t-distribution includes both the skew normal and the normaldistributions as special cases. Inference for the skew t-model becomes problematicin these cases because the expected information matrix is singular and the parametercorresponding to the degrees of freedom takes a value at the boundary of its parameterspace. In particular, the distributions of the likelihood ratio statistics for testing thenull hypotheses of skew normality and normality are not asymptotically χ2. Theasymptotic distributions of the likelihood ratio statistics are considered by applying theresults of Self and Liang (1987) for boundary-parameterinference in terms of reparameterizations designed to remove the singularity of theinformation matrix. The Self–Liang asymptotic distributions are mixtures, and it isshown that their accuracy can be improved substantially by correcting the mixingprobabilities. Furthermore, although the asymptotic distributions are non-standard,versions of Bartlett correction are developed that afford additional accuracy. Bootstrapprocedures for estimating the mixing probabilities and the Bartlett adjustment factorsare shown to produce excellent approximations, even for small sample sizes.