Bayes inference for treatment effects with uncertain order constraints

Constrained parameter situations arise in a wide variety of practical problems and the corresponding order restricted inference has been extensively researched. In previous work, order restrictions have always been imposed on the estimates of the ordered parameters, the data may, however, provide strong evidence that the constraints are untrue, in which case it might be more sensible for the estimates to contradict the constraints, or to compromise between unconstrained estimates and estimates based on the constraint. In this paper, we consider finite sample inference for the one-way layout normal means problem with unknown common variance and we assume that the treatment means are hypothesized to be ordered but with a degree of uncertainty in this hypothesis via prior assumptions that we express. This flexibility will permit the data to play a more substantive role in the inferential procedure. The posterior distribution of the treatment means is estimated using the Gibbs sampler. An illustrative analysis using a real data set is provided.