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= Aµ where A
is an l× k 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.
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