Estimation of scatter matrix based on i.i.d. sample from elliptical distributions

In this paper, we consider the estimation of a scatter matrix under entropy loss, quadratic loss, when the samples x₍₁₎,... x_(n) are i.i.d. and x₍₁₎∼EC_p(μ,Σ,f). With respect to entropy and quadratic losses, we obtain the best estimator of Σ having the form αS_x as well as having the form T_xΔT_x′, where S_x,T_x and Δ are given in the text, and obtain the minimax estimator of Σ and the best equivariant estimator of Σ with respect to the triangular transformations group LT⁺(p) (the group consisting of lower triangular matrices with positive diagonal elements). Some related discussion are given as its generalizations.