Further characterizations of design optimality and admissibility for partial parameter estimation in linear regression
The problem of finding optimal linear regression designs for the estimation of \(s<k\) linear functions Ka (where K is an \(s\times k\) matrix) of the k regression parameters \(a=(a_ 1,...,a_ k)^ t\) is considered. The optimality criteria \(\phi\) considered are real-valued functions on the set PD(s) of all positive definite \(s\times s\) matrices which are convex and decreasing. An (approximate) design \(\xi_ 0\) is \(\phi\)-optimal for this problem iff Ka is estimable under \(\xi_ 0\) and \(\xi_ 0\) minimizes \(\phi\) (J(\(\xi)\)) over the set of all \(\xi\) under which Ka is estimable, J(\(\xi)\) being the reduced information matrix of \(\xi\) for Ka. A new equivalence theorem for this minimization problem is presented and the results are strengthened for invariant designs. Design admissibility for the estimation of Ka is discussed and a necessary condition and a sufficient condition for admissibility are given with special focus on invariant designs. Examples are given to illustrate possible applications of the results.
scientific article; zbMATH DE number 4060569