Simultaneous selection and estimation of the best treatment with respect to a control in a general linear model

The problem of selecting the largest treatment parameter, provided it is better than a control, and simultaneously estimating the selected treatment parameter in a general linear model is considered in the decision theoretic Bayes approach. Both cases, where the error variance is known or unknown, are included. Bayes decision rules are derived for noninformative and for normal priors. Bayes rules for noninformative priors are derived under a general loss function for designs that satisfy the BTIB condition of Bechhofer and Tamhane (Technometrics 23 (1981) 45). For unbalanced designs, a linear loss function is adopted and it is demonstrated, via simulations, that the simultaneous estimation of the selected treatment effect plays an important role in correcting an undesirable effect for the selection problem.