Linda M. Haines

The application of the annealing algorithm to the construction of exact optimal designs for linear-regression models

This article reports the application of the annealing algorithm to the construction of exact D−, I−, and G-optimal designs for polynomial regression of degree 5 on the interval [—1, l] and for the second-order model in two factors on the design space [—1, l] × [—1, 1]. Details of the perturbation scheme and the annealing schedules used are given, and the method of implementation is illustrated by means of a simple example. The algorithm is assessed by comparing its performance, in terms of computer time and effkiency, with the modified Fedorov procedure, and it is shown to be particularly effective in finding G-optimal designs. The salient features of the exact designs constructed in this study are also summarized.

A statistical approach to the analytic hierarchy process with interval judgements. (I). Distributions on feasible regions

Abstract This paper addresses the problem of extracting preferences for alternatives from interval judgement matrices in the Analytic Hierarchy Process (AHP). The method of Arbel for extracting such preferences, which is based on the assumption that the interval judgements specify a feasible region in the weight space of the alternatives, is critically appraised from a statistical perspective and some new ideas emanating from this approach are developed and discussed. In particular it is proposed that a distribution for the weights on the feasible region, which is both tractable and meaningful, be adopted. The mean of the distribution can then be used as an assessment of the overall ranking of the alternatives and quantities of interest, such as probabilities and marginal distributions, can immediately be quantified. Two specific distributions on the feasible region, the uniform distribution and the distribution of random convex combinations with coefficients which are uniform spacings, are examined in some detail and the ideas which emerge are illustrated by means of selected examples.

Bayesian D-optimal designs for the exponential growth model

Abstract Bayesian optimal designs for nonlinear regression models are of some interest and importance in the statistical literature. Numerical methods for their construction are well-established, but very few analytical studies have been reported. In this paper, we consider an exponential growth model used extensively in the modelling of simple organisms, and examine the explicit form of the Bayesian D-optimal designs. In particular, we show that Dθ-optimal designs for this model are balanced two-point designs for all values of the parameters. We further derive explicit expressions for Bayesian D-optimal designs which are based on exactly two points of support, and provide necessary and sufficient conditions for such designs to exist. We illustrate our results by means of two examples.

Optimal design for nonlinear regression models

Two results for D θ-optimal designs for nonlinear regression models are shown to follow directly from approximate design theory. The first result considered is one concerning the replication of exact designs with minimum support, first established by Atkinson and Hunter and by M.J. Box in 1968, while the second pertains to a heteroscedastic model introduced by Velilla and Llosa in 1992. An illustrative example is provided.

Nonlinear models for neural networks

The statistical principles underpinning hidden-layer feed-forward neural networks for fitting smooth curves to regression data are explained and used as a basis for developing likelihood- and bootstrap-based methods for obtaining confidence intervals for predicted outputs.

Interval Judgements in the Analytic Hierarchy Process: A Statistical Perspective

Two broad approaches are commonly used in the Analytic Hierarchy Process for deriving a suitable ranking of alternatives from an interval judgement matrix. In the present study these approaches are examined from a statistical perspective and are extended and developed accordingly. The ideas are introduced and illustrated by means of a simple example.