Bayesian optimal designs for linear regression models

Sadi Mahmoud El-krunz, Purdue University

Abstract

Bayesian optimal designs for estimation and prediction in linear regression models are considered. For a large class of optimality criteria $\Phi$, a general Kiefer's type equivalence theorem is proved. This equivalence theorem is used to derive a Bayesian version of Elfving's Theorem for the c-optimality criterion and the class of prior precision matrices R for which the Bayesian c-optimal designs are supported by the points of the classical c-optimal design is characterized. It is also proved that the Bayesian c-optimal design, for n large enough, is always supported at the same support points of the classical c-optimal design $\xi\sp\*$ if $\xi\sp\*$ is supported at exactly k distinct points and for a large class of prior precision matrices R if $\xi\sp\*$ is supported at 1 $\leq$ m $<$ k points. Duality theory is then used to give another proof of Elfving's Theorem for Bayesian c-optimality and conditions under which a one point design is Bayesian c-optimum are given. Emphasis is laid on the geometry inherent in the Bayesian c-optimal design problem and the parallelism between classical and Bayesian c-optimal design theory is illustrated. A number of examples are given to illustrate how Elfving's Theorem can be used to construct Bayesian c-optimal designs. The geometry - duality approach is extended for the $\Psi$-optimality criterion and a matrix analog of the geometric result of Elfving is derived and its applications are discussed.

Degree

Ph.D.

Advisors

Studden, Purdue University.

Subject Area

Statistics

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