Minimal efficiency of designs under the class of orthogonally invariant information criteria

Consider the linear regression model with uncorrelated errors and an experimental design ξ. In the article, we address the problem of calculating the minimal efficiency of ξ with respect to the class [InlineMediaObject not available: see fulltext.] of orthogonally invariant information criteria, containing all Kiefer’s criteria of ϕ<Subscript> p </Subscript>-optimality, among others. We show that the [InlineMediaObject not available: see fulltext.]-minimal efficiency of ξ is equal to the minimal efficiency of ξ with respect to a finite class of criteria which generalize the criterion of E-optimality. We also formulate conditions under which a design is maximin efficient, i.e. the most efficiency-stable for criteria from [InlineMediaObject not available: see fulltext.]. To illustrate the results, we calculated the [InlineMediaObject not available: see fulltext.]-minimal efficiency of ϕ<Subscript> p </Subscript> (in particular D, A and E) optimal designs for polynomial regression on [−1,1] up to degree 4. Moreover, for the quadratic model we explicitly constructed the [InlineMediaObject not available: see fulltext.]-maximin efficient design. Copyright Springer-Verlag 2004