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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.

##### MSC:
 62K05 Optimal statistical designs 62J05 Linear regression; mixed models
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