Testing for Normality When the Sampled Distribution Is Extended Skew-Normal

The extended skew-normal (ESN) distribution includes the normal one as a special case. Unfortunately, its information matrix is singular under normality, thus preventing application of standard likelihood-based methods for testing the null hypothesis of normality. The paper shows that for univariate ESN distributions the sample skewness provides the locally most powerful test for normality among the location and scale invariant ones. The generalization to multivariate ESN distributions considers projections of the data onto the direction corresponding to maximal skewness. Related computational problems simplifies for the multivariate ESN distribution, where the direction maximizing skewness is shown to have a simple parametric interpretation.