Minimax estimation of the mean of spherically symmetric distributions under general quadratic loss

AbstractFor X one observation on a p-dimensional (p ≥ 4) spherically symmetric (s.s.) distribution about θ, minimax estimators whose risks dominate the risk of X (the best invariant procedure) are found with respect to general quadratic loss, L(δ, θ) = (δ − θ)′ D(δ − θ) where D is a known p × p positive definite matrix. For C a p × p known positive definite matrix, conditions are given under which estimators of the form δa,r,C,D(X) = (I − (ar(|X|2)) D−12CD12 |X|−2)X are minimax with smaller risk than X. For the problem of estimating the mean when n observations X1, X2, …, Xn are taken on a p-dimensional s.s. distribution about θ, any spherically symmetric translation invariant estimator, δ(X1, X2, …, Xn), with have a s.s. distribution about θ. Among the estimators which have these properties are best invariant estimators, sample means and maximum likelihood estimators. Moreover, under certain conditions, improved robust estimators can be found