A Monte Carlo Comparison of Tests for Multivariate Normality Based on Multivariate Skewness and Kurtosis.

Abstract

The assumption of multivariate normality (MVN) underlies many common parametric multivariate statistical procedures, and numerous tests have been defined for testing the assumption. Among these tests, those based on concepts of "multivariate skewness" and "multivariate kurtosis" hold special interest since they appear to test for specific types of departures from MVN. This research uses Monte Carlo simulation to compare the performance of several MVN tests which are based on various definitions of multivariate skewness and kurtosis. Specifically, the tests are Mardia's (1970) b$\sb{\rm 1,p}$ and b$\sb{\rm 2,p}$, Small's (1980) Q$\sb1$ and Q$\sb2$, and Srivastava's (1984) b$\sb{\rm 1p}$ and b$\sb{\rm 2p}$. Two main issues are addressed. First, Mardia's tests are affine invariant, while those of Small and Srivastava are coordinate dependent. Conjectures are advanced regarding the conditions under which coordinate-dependent tests will perform better than affine-invariant tests and vice versa. A Monte Carlo experiment is constructed to evaluate these conjectures. It is concluded that neither coordinate-dependent nor affine-invariant tests can be eliminated from consideration, since each type is strongly superior to the other under certain circumstances. These circumstances pertain to whether or not those third- and fourth-order moments involving more than one variable in the coordinate system have normal or non-normal values. The second issue concerns the distributional dependency of skewness tests. It is conjectured, in particular, that skewness tests based on third-order moments (which includes all skewness tests considered here) are highly distributionally dependent, with this dependency being related to the same distributional characteristic that determines kurtosis. It is further conjectured that this dependency remains of importance asymptotically A Monte Carlo experiment is designed to evaluate these conjectures. Results confirm the dependency and that it is not simply a small sample problem. Based on this, it is concluded that "skewness" tests are not truly diagnostic; that is, they do not distinguish well between "skewed" and "non-skewed" distributions. In particular, skewness tests are likely to identify as "skewed" many non-skewed distributions with greater than MVN kurtosis; and they will fail to identify as "skewed" many skewed distributions with less than MVN kurtosis.