ESTIMATION OF THE LARGER NORMAL MEAN

Let X and Y be sample means drawn from two independent normal distributions with unknown mean μ_i and known variance σ^2_i for i = 1, 2, respectively, and let μ^* ≡ max(μ_1, μ_2). The purpose of the present paper is to propose a new estimator of μ^* as a hybrid estimator η(C) with a given non-negative C, which includes both estimators, an estimator η_1 of a linear combination of X and Y and a simple estimator η_2 = max(X, Y). After comparing η(C) numerically with the maximum likelihood estimator δ given by Blumenthal and Cohen (1968b), it is found that η(C) is better than δ for C ∈ [1.87, 1.91] in view of the mean square error. The C has also been investigated in the sense of minimax regret, and an optimal C is obtained numerically as C^* = 1.704.