Matrix equivalence classes with applications Original Research Article

A definite matrix Σ is represented in terms of a semidefinite matrix Ω and a vector γ, such that (Ω,γ) may be recovered explicitly from Σ. Equivalence classes are generated over the cone of positive definite matrices, first on fixing Ω, then on fixing γ. These classes in turn associate naturally as scale parameters in ensembles of elliptical measures. Each ensemble is seen to admit a distinct linear solution to the problem of minimal risk equivariant estimation for a location scalar, as do all mixtures over a particular ensemble to include star-contoured measures.