The asymptotic behavior of a variant of multivariate kurtosis
Let be independent identically distributed random d-dimensional column vectors with arithmetic mean [Xbar] n and empirical covariance matrix S n. Apart from the celebrated kurtosis measure of Mardia, there has been recent interest in the variant which formally constitutes a closer analogue to the multivariate skewness measure , than b2,d . We show that, under certain moment restrictions and the weak assumption that the support of the underlying distribution has positive Lebesgue measure, the asymptotic distribution of , suitably normalized, is normal. Moreover, the joint limiting distribution of b2,d and is bivariate normal. Within the class of elliptically symmetric distributions the asymptotic correlation between b2,d and is 1. The consistency class of a test for multivariate normality based on is determined.