Testing for submodels in the context of the skew-t distribution

The skew $t$-distribution includes both the skew normal and the normal distributions as special cases. Inference for the skew $t$-model becomes problematic in these cases because the expected information matrix is singular and the parameter corresponding to the degrees of freedom takes a value at the boundary of its parameter space. In particular, the distributions of the likelihood ratio statistics for testing the null hypotheses of skew normality and normality are not asymptotically chi-squared. The asymptotic distributions of the likelihood ratio statistics are considered by applying the results of Self and Liang (1987) for boundary-parameter inference in terms of reparameterizations designed to remove the singularity of the information matrix. The Self-Liang asymptotic distributions are mixtures, and it is shown that their accuracy can be improved substantially by correcting the mixing probabilities. Furthermore, although the asymptotic distributions are non-standard, versions of Bartlett correction are developed that afford additional accuracy. Bootstrap procedures for estimating the mixing probabilities and the Bartlett adjustment factors are shown to produce excellent approximations, even for small sample sizes.