A comparison of tests for multivariate normality that are based on measures of multivariate skewness and kurtosis

The examination of coefficients of multivariate skewness and kurtosis is one of the more commonly used techniques for assessing multivariate normality (MVN). In this article, several tests for MVN based on these coefficients are compared via Monte Carlo simulation. The tests considered here include those based on Mardia's affine-invariant measures of multivariate skewness and kurtosis and an omnibus procedure that combines the two. Also included are Small's tests, which are based on combinations of the marginal skewness and kurtosis coefficients and are coordinate-dependent. These tests are compared in terms of their power against a wide variety of non-MVN distributions; included in these alternatives are distributions with iid components, as well as distributions with positively-correlated components. Among the alternatives considered are non-MVN distributions with skewed components, symmetric components, univariate normal components, and MVN values of skewness and kurtosis. The tests considered here perform as expected; the affine-invariant procedures outperform the coordinate-de¬pendent ones when the variables are correlated, whereas the coordinate-dependent tests are more powerful when the variables are uncorrelated. Neither type of procedure performs well when the alternative distribution has MVN values of skewness and kurtosis. The practical implications of these results for assessing MVN, as well as a method for combining the two types of tests, are discussed. The limitations of the use of coefficients of multivariate skewness and kurtosis in “diagnosing” departures from MVN are also described.