Valery V. Fedorov

Invited Discussion Paper Constrained Optimization of Experimental Design

This is an attempt to discuss various approaches developed in experimental design when constraints are imposed. These constraints may be on the total cost of the experiment, the location of the supporting point, the value of auxiliary objective functions, and so on. The basic idea of the paper is that all corresponding optimization problems can be imbedded in the convex theory of experimental design. Part 1 is concerned with the properties of optimal designs, while Part 2 is devoted mainly to numerical methods. We have tried to avoid details, emphasizing ideas rather than technicalities. This is not intended as a literature review. The authors subjectively surely left many excellent papers behind.

Minimum bias designs with constraints

A new class of model-robust optimality criteria, based on the mean squared error, is introduced in this paper. The motivation is to find designs when the researcher is more concerned with controlling the variance than the bias, or vice versa. The set of criteria proposed here is also appealing from a mathematical perspective in the sense that, unlike the Box and Draper (1959, J. Amer. Statist. Assoc. 54, 622–654), criterion, they can be imbedded in the framework of convex design theory and, hence, facilitate the search for globally optimal designs. The basic idea is to minimize a convex function of the bias part of the mean squared error subject to a convex constraint on the variance part, or vice versa. Equivalence theorems are derived and examples for the linear and quadratic regression problems are provided.

Another view on optimal design for estimating the point of extremum in quadratic regression

In this paper we illustrate how certain design problems can be simplified by reparametrization of the response function. This alternative viewpoint provides further insights than the more traditional approaches, like minimax, Bayesian or sequential techniques. It will also improve a practitioner’s understanding of more general situations and their “classical” treatment.

Moving local regression: The weight function

Moving local regression is a nonparametric technique for smoothing, interpolating and forecasting by means of locally fitted regression models. The paper explores the “optimal” structure of the weight function, taking into account the location of supporting points and the suspected behaviour of the remainder term, and surveys results or choice of weight functions in traditional moving local regression approaches.

Optimal design and the model validity range

Abstract A class of model-robust optimal designs, based on an extension of the standard optimality criteria to cases where there exist some prior information on the validity of a response function, is considered. Under this set-up, the concept of the “model validity range” is introduced and explored. A necessary condition for optimality is obtained for the determinant criterion ( D -optimality in the classical case). A modified version of this criterion is proposed and discussed. The corresponding results provide upper and low bounds for the original problem and help to construct approximate solutions, when contamination is relatively small. Optimal designs for simple but commonly used regression models are obtained and studied.

Optimal Designs for Time—Dependent Responses

The purpose of this paper is to investigate methods for the design of optimal dynamic experiments. In Section 2, we introduce notation and a determinant criterion for dynamic experiments. In Section 3, optimal dynamic designs are constructed analytically in a number of simple cases. As will be seen, the dimension of the design problem increases with the dimension of the control vectors (equivalently, the response vector). In Section 4, we discuss the use of suitable parameterizations of the control variable trajectories to reduce the dimensionality of the optimization problem. The relationship between the design of optimal dynamic experiments and the results developed for marginally restricted designs is considered in Section 5. Numerical methods are discussed in Section 6. We close, in Section 7, with an extension of the methodology to the case where one or more linear combinations of the response trajectory is observed for each control trajectory.

Optimized Local Regression: Computational Methods for the Moving Average Case

Optimized moving local regression is an extension of Cleveland’s loess technique that takes a suspected misspecification of the model into account. The weights are chosen so that the effect of the misspecification is minimized. The derivation of optimal weights is shown to be similar to that of an optimal design for an experiment.