Robust Bayesian experimental design and estimation for analysis of variance models using a class of normal mixtures

Abstract

Abstract In this paper we address the problem of Bayesian experimental design and estimation for ANOVA models when the prior distribution belongs to a class of finite mixtures of normal distributions. We introduce new optimality criteria and derive an estimator and the corresponding experimental design which simultaneously are optimal for the class. The proposed estimator minimizes an average posterior expected loss over the class of posterior distributions, the experimental design minimizes an average of the Bayes risks of the proposed estimator over this class. The optimization of this criterion is mathematically tractable, allowing us to give closed form solutions for the estimation problem even in more complex ANOVA models such as two-way ANOVA and block designs. For the one-way ANOVA model we also derive an alternative estimator and the corresponding design according to a minimax-regret criterion. The similar performance of these two procedures is illustrated in an example.