Design and inference in nonlinear problems

Abstract

The aim of this thesis is to review and augment the theory and methods of optimal nonlinear experimental design. It represents a continuation of the work on experimental design in the Department of Statistics in Glasgow University (Silvey and Titterington (1973), Ford (1976), Silvey (1980), Titterington (1980a,b), Ford and Silvey (1980), Torsney (1981) among others). Chapter 1 serves as an introduction to the nonlinear problem. In Chapter 2 we formulate the appropriate notation needed for the development of this thesis. The main assumptions, which we will recall if needed, and the necessary theory is discussed. In Chapter 3 the idea of the optimal nonlinear experimental design is formulated for any convex criterion function. This leads to traditional definitions as special cases. We also focus on the canonical form of a design under c(e)-optimality. Partially nonlinear models are reviewed and the design for a subset of parameters is discussed in the context of the general optimality criterion function. The geometrical aspects of the nonlinear case are compared and contrasted with the linear case. Chapters 4 and 5 are devoted to strategies for the construction of the nonlinear optimal designs. Alternative approaches for the static design problem are discussed in Chapter 4. Emphasis is given to the sequential approach to design in Chapter 5. There, binary response problems are also tackled and the stochastic approximation method is reviewed and discussed. Chapter 6 is devoted to confidence intervals. The problem of constructing confidence Intervals if the sequential principle of design is adopted is discussed and a suggestion is given. As a result a simulation study is presented. In Chapter 7 two more simulation studies are analysed, the first for a one parameter binary problem and the second for a two parameter regression problem. Different designing procedures are applied and more emphasis is given to sequential methods. The stochastic approximation method is discussed as a fully sequential method. The performance of approximate confidence intervals is investigated. Chapter 8 considers a compromise between the static and fully sequential design. The calibration problem is used as an example and investigated in a (yet another) simulation study. The maxi-min efficiency design is derived and investigated. In Chapter 9 we examine a design problem in rhythmometry involving the cosinor function. Different design criteria are Introduced for the full sample space as well as a truncated form. Geometrical ideas provide a solution to solve this problem. An analytical approach is also offered as a method of solution of this practical design problem.