On Fisher information matrices and profile log-likelihood functions in generalized skew-elliptical models

In recent years, the skew-normal models introduced in Azzalini (1985) have enjoyed an amazing success, although an important literature has reported that they exhibit, in the vicinity of symmetry, singular Fisher information matrices and stationary points in the profile log-likelihood function for skewness, with the usual unpleasant consequences for inference. For general multivariate skew-symmetric and skew-elliptical models, the open problem of determining which symmetric kernels lead to each such singularity has been solved in Ley and Paindaveine (2010). In the present paper, we provide a simple proof that, in generalized skew-elliptical models involving the same skewing scheme as in the skew-normal distributions, Fisher information matrices, in the vicinity of symmetry, are singular for Gaussian kernels only. Then we show that if the profile log-likelihood function for skewness always has a point of inflection in the vicinity of symmetry, the generalized skew-elliptical distribution considered is actually skew-(multi)normal. In addition, we show that the class of multivariate skew-t distributions (as defined in Azzalini and Capitanio 2003), which was not covered by Ley and Paindaveine (2010), does not suffer from singular Fisher information matrices in the vicinity of symmetry. Finally, we briefly discuss the implications of our results on inference.