Optimal Estimation under Linear Constraints

This paper shows how a multivariate Bayes estimator can be adjusted to satisfy a set of linear constraints. In the direct approach, the constraint is enforced by a restriction on the class of admissible estimators. In an alternative approach, the constraint is merely encouraged by a mixed risk function which penalises misbalance between the estimator and the constraint. The adjustment to the optimal unconstrained estimator is shown to depend on the risk function and the linear constraints only, not on the probability model underlying the Bayes estimator. Two practical examples are given, one of which involves reconciliation of independently assessed share values with current market values.