A unified approach to non-minimaxity of sets of linear combinations of restricted location estimators

This paper studies minimaxity of estimators of a set of linear combinations of location parameters μi, i=1,...,k under quadratic loss. When each location parameter is known to be positive, previous results about minimaxity or non-minimaxity are extended from the case of estimating a single linear combination, to estimating any number of linear combinations. Necessary and/or sufficient conditions for minimaxity of general estimators are derived. Particular attention is paid to the generalized Bayes estimator with respect to the uniform distribution and to the truncated version of the unbiased estimator (which is the maximum likelihood estimator for symmetric unimodal distributions). A necessary and sufficient condition for minimaxity of the uniform prior generalized Bayes estimator is particularly simple. If one estimates θ=Atμ where A is a k×ℓ known matrix, the estimator is minimax if and only if (AAt)ij≤0 for any i and j (i≠j). This condition is also sufficient (but not necessary) for minimaxity of the MLE.