Statistical theory of shape under elliptical models and singular value decompositions

The size-and-shape and shape distributions based on non-central and non-isotropic elliptical distributions are derived in this paper by using the singular value decomposition (SVD). The general densities require the computation of new integrals involving zonal polynomials. The invariance of the central shape distribution is also proved. Finally, some particular densities are applied in a classical data of Biology, and the inference based on exact distributions is performed after choosing the best model by using a modified BIC∗BIC∗ criterion.

► Shape theory is studied under the family of non-isotropical elliptical distributions.
► New integrals involving zonal polynomials are derived.
► Exact inference of shape theory based on a detailed isotropic case is performed.