  Elliptically contoured distributions can be considered to be the
distributions for which the contours of the density functions are proportional
ellipsoids. Kamiya, Takemura and Kuriki (2006) generalized the elliptically
contoured distributions to star-shaped distributions, for which the contours
are allowed to be arbitrary proportional star-shaped sets. This was achieved by
considering the so-called orbital decomposition of the sample space in the
general framework of group invariance. In the present paper, we extend their
results by conducting the orbital decompositions in steps and obtaining a
further, hierarchical decomposition of the sample space. This allows us to
construct probability models and distributions with further independence
structures. The general results are applied to the star-shaped distributions
with a certain symmetric structure, the distributions related to the two-sample
Wishart problem and the distributions of preference rankings.
