OPTIMAL RESTRICTED EXPERIMENTAL DESIGNS

KENNETH BRUCE CONSTANTINE, Purdue University

Abstract

The problem of designing an experiment to obtain uncorrelated data from the linear model y(x) = (theta)'f(x)+ (epsilon) is considered under a restriction on the allocation of observations. It is assumed that such a restriction requires that an approximate design (xi) on (chi) belongs to some closed convex set (XI) of probability measures. Particular types of design restrictions investigated include: (1) no observations may be allocated to some open set (chi)-E; (2) limiting measures (phi) and (psi) on (chi) are prescribed and d(phi) (LESSTHEQ) d(xi) (LESSTHEQ) d(psi); (3) the proportions of observations allocated to certain open sets are limited; and (4) (chi) is a product space and all designs must have a prescribed marginal distribution (xi)(,1)('*). The admissible designs for (univariate) polynomial regression are characterized for restrictions (1) and (2). A necessary condition for admissibility is provided for restriction (3). This condition is shown to be sufficient in two special cases of (3). For each type of restriction, the admissibility condition is expressed in terms of a suitably defined index I((xi)) which appropriately "counts" the support points of a design. Secondary consideration is given to methods for improving upon an inadmissible design. A characterization of D-optimality is provided which adapts the Kiefer and Wolfowitz ('60) equivalence result to the restricted design setting. Several applications to restrictions (1) - (4) are given. The optimal designs obtained are compared with one another and the unrestricted solutions. A characterization of c-optimality is given which generalizes a result of Kiefer and Wolfowitz ('59) to the restricted design setting. Applications to restrictions (1) - (4) are provided. These include the (implicit) solution to a problem of Hoel ('65).

Degree

Ph.D.

Subject Area

Statistics

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