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Minimal efficiency of designs under the class of orthogonally invariant information criteria. (English) Zbl 1079.62072
Summary: Consider a linear regression model with uncorrelated errors and an experimental design \(\xi\). We address the problem of calculating the minimal efficiency of \(\xi\) with respect to the class \(\mathbb O\) of orthogonally invariant information criteria, containing all Kiefer criteria of \(\phi_{p}\)-optimality, among others. We show that the \(\mathbb O\)-minimal efficiency of \(\xi\) is equal to the minimal efficiency of \(\xi\) 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 \(\mathbb O\). To illustrate the results, we calculated the \(\mathbb O\)-minimal efficiency of \(\phi_{p}\) (in particular \(D\), \(A\) and \(E\)) optimal designs for polynomial regressions on \([-1,1]\) up to degree 4. Moreover, for the quadratic model we explicitly constructed the \(\mathbb O\)-maximin efficient design.

62K05 Optimal statistical designs
62J05 Linear regression; mixed models
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