On the Problem of More Than One Kurtosis Parameter in Multivariate Analysis

When a multivariate elliptical distribution is used as the basis in multivariate analysis all fourth-order cumulants are expressed in terms of a single kurtosis parameter. This and other well-known properties place unrealistic restrictions on the distribution of the covariance matrix. In this paper a class of elliptical distributions that can be expanded as a power series is first defined. An effort is then made to introduce meaningful multivariate distributions that are related to these elliptical distributions and that contain more than one kurtosis parameter.