96-F-15. Takemura, Akimichi and Kuriki Satoshi, "Theory of Cross Sectionally Contoured Distributions and Its Applications", July 1996.

We discuss generalization of elliptically contoured distributions to densities whose contours are arbitrary cross sections in the framework of group invariance. This generalization leads to much richer family of distributions compared to the elliptically contoured distributions. The basic property of the elliptically contoured distribution is the independence of the `length' and the `direction' of the random vector. We show that in our generalized framework, this independence still holds if we define the length appropriately. Our examples include `star-shaped distributions' and their generalization to random matrices.