Shapiro-Wilk test for multivariate skew-normality

The multivariate skew-normal family of distributions is a flexible class of probability models that includes the multivariate normal distribution as a special case. Two procedures for testing that a multivariate random sample comes from the multivariate skew-normal distribution are proposed here based on the estimated canonical form. Canonical data are transformed into approximately multivariate normal observations and then a multivariate version of the Shapiro-Wilk test is used for testing multivariate normality. Critical values for the tests are approximated without using parametric bootstrap. Monte Carlo simulation results provide evidence that the nominal test level is preserved, in general, under the considered settings. The simulation results also indicate that these tests are in general more powerful than existing tests for the same problem versus the studied alternatives.