On weighted optimality of experimental designs

TL;DR

This paper establishes the equivalence between weighted and standard optimality criteria in experimental design, extends the theory to general estimable functions, and provides practical methods for constructing weight matrices.

Contribution

It proves the equivalence of weighted and standard optimality criteria, extends weighted optimality to any estimable functions, and offers a simple, interpretable weight matrix construction.

Findings

Weighted optimality is equivalent to standard optimality for estimable functions.

A systematic method to construct weight matrices with clear interpretation.

Extension of weighted optimality theory to broader classes of estimable functions.

Abstract

When the experimental objective is expressed by a set of estimable functions, and any eigenvalue-based optimality criterion is selected, we prove the equivalence of the recently introduced weighted optimality and the 'standard' optimality criteria for estimating this set of functions of interest. Also, given a weighted eigenvalue-based criterion, we construct a system of estimable functions, so that the optimality for estimating this system of functions is equivalent to the weighted optimality. This allows one to use the large body of existing theoretical and computational results for the standard optimality criteria for estimating a system of interest to derive theorems and numerical algorithms for the weighted optimality of experimental designs. Moreover, we extend the theory of weighted optimality so that it captures the experimental objective consisting of any system of estimable functions, which was not the case in the literature on weighted optimality so far. For any set of estimable functions, we propose a corresponding weight matrix of a simple form, and with a straightforward interpretation. Given a set of estimable functions with their corresponding weights, we show that it is useful to distinguish between the primary weights selected by the experimenters and the secondary weights implied by the weight matrix.