Some Remarks on Koziol's Kurtosis

James Koziol, in his 1987 and 1989 papers, proposed to use the sums of either the squared fourth-order cumulants or moments as test statistics for multivariate normality. His proposals are by far less popular than Mardia's measure of multivariate kurtosis, that is the fourth moment of the Mahalanobis distance of a random vector from its mean. We investigate some properties of Koziol's measures of multivariate kurtosis which motivate their use in statistical practice. Firstly, we show some of their connections with Mahalanobis angles. Secondly, we use inequalities to highlight their connections with other measures of multivariate skewness and kurtosis. Thirdly, we obtain their analytical formulae for some well-known multivariate statistical models. Simple examples illustrate the interpretation of Koziol's measures of multivariate kurtosis and detect a wrong statement about them which appeared in the statistical literature. We suggest that Mardia's and Koziol's measures of kurtosis should be used together to detect interesting data structures.