Singular Information Matrices, Directional Derivatives, and Subgradients in Optimal Design Theory

Summary

A linear regression setup is considered, where the parameters of interest are only part of or a linear function of the full set of unknown parameters. Under the standard statistical assumptions the problem of finding an optimal approximate design leads to a convex minimization problem of the following type:

Minimize Φ(M) = φ(J(M)) over the set of all information matrices M of those approximate designs which allow a linear unbiased estimator for a given ℝ^S — valued linear function of the unknown parameters.

The optimality criterion φ is a convex and decreasing real-valued function on the set of all positive definite (s × s)-matrices, and J(M) is the reduced information matrix of M. The possibility of an optimal design with a singular M has caused a great many difficulties when trying to obtain characterizations for optimality via directional derivatives as in the “regular case”. Recently the author has given a description of directional derivatives by means of shorted matrices and minimum seminorm generalized inverses. The present paper gives some further results. The set of all subgradients of the objective function Φ is found. As a result, a general equivalence theorem for optimality can be derived directly using a major theorem of convex analysis, as pointed out by Pukelsheim and Titterington (1983). A saddle point, characterization for optimal designs is proved which has a nice interpretation in terms of a least favourable transformation of the linear regression setup.