Quadratic Loss of Order Restricted Estimators for Treatment Means with a Control

We consider an experiment which consists of k treatment groups and a control group. Let the sample means Y₀, Y₁, , Yₖ be independent normal variates with expected values ₀, ₁, , ₖ and with variances ²/n₀, ²/n₁, , ²/nₖ. Let w₀, w₁, , wₖ be positive weights and let ^₀, ^₁, , ^ₖ be the weighted least squares estimators subject to the constraints ₀ ᵢ, i = 1, , k. We establish that for large k, E (^₀ - ₀) ² > E (Y₀ - ₀) ² when wᵢ = nᵢ, i = 0, 1, , k. Under suitable conditions, we show that E (^ᵢ - ᵢ) ² < E (Yᵢ - ᵢ) ², i = 0, 1, , k.