Results in the Theory and Construction of D-Optimum Experimental Designs

Abstract

Summary A method of generating D‐optimum design measures as defined by Kiefer and Wolfowitz is briefly described. The relevant theorem is generalized to yield a method of generating Ds‐optimum design measures for a specific subset of s of the parameters in the regression model. The one parameter case is described in further detail. An analogous theorem for discrete designs with a fixed number of points is proved giving a lower bound to the optimality achievable with such designs. A final section is devoted to the loss in efficiency entailed by almost equal replication of a given discrete design. The discussion is extended to the one‐dimensional polynomial model and the optimality of the replication is conjectured for the general case and proved for the quadratic case.

A Comparison of Algorithms for Constructing Exact D-Optimal Designs

“…The Fedorov algorithm discussed here is distinct from Fedorov's well-known algorithm for generating approximate Doptimal designs. The second (henceforth referred to as the Wynn-Mitchell algorithm) is due independently to Mitchell and Miller (1970) and to Wynn (1972). Both the Fedorov and Wynn-Mitchell algorithms are single point exchange algorithms.…”

Fast algorithms for the minimum volume estimator

“…It is proved in [19] that MVEE of the original set is the intersection of the MVEE of the centralized set with the hyperplane x n+1 = 1. Problem (D) is the well-known statistical problem of finding a D-optimal design measure on the set X (See [5,6,9,24,27,36,37]). A detailed proof of the duality of the two problems above can be found in [2].…”

“…Various such algorithms were proposed and analysed by statisticians since the 1960s. The first algorithms appeared in [9] and [37], which only considered iterations that increase the weight of a chosen coordinate. These methods were very slow, and soon were improved by [6] by allowing the weight of the chosen coordinate to also decrease, referred often as the away steps.…”

Combined algorithm to compute D-optimal designs

“…The introduction to the mentioned algorithms was pioneered by Wynn [12,13], and Fedorov [1] and since then, the construction of such algorithm has tried to adapt for more general criteria (Fedorov and Malyutov [14], Whittle [15], Gribik and Kortanek [16] and Atwood [17][18][19]). This point encouraged the development of the general equivalence theorems (see, for example, Kiefer [20]).…”