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

Abstract

We consider an experiment which consists of $k$ treatment groups and a control group. Let the sample means $\bar{Y}_0, \bar{Y}_1, \cdots, \bar{Y}_k$ be independent normal variates with expected values $\mu_0, \mu_1, \cdots, \mu_k$ and with variances $\sigma^2/n_0, \sigma^2/n_1, \cdots, \sigma^2/n_k$. Let $w_0, w_1, \cdots, w_k$ be positive weights and let $\mu^\ast_0, \mu^\ast_1, \cdots, \mu^\ast_k$ be the weighted least squares estimators subject to the constraints $\mu_0 \leq \mu_i, i = 1, \cdots, k$. We establish that for large $k, E(\mu^\ast_0 - \mu_0)^2 > E(\bar{Y}_0 - \mu_0)^2$ when $w_i = n_i, i = 0, 1, \cdots, k$. Under suitable conditions, we show that $E(\mu^\ast_i - \mu_i)^2 < E(\bar{Y}_i - \mu_i)^2, i = 0, 1, \cdots, k$.