Title:

Construction of optimising distributions with applications in estimation and optimal design

This thesis considers constructing optimising distributions with applications in estimation and optimal design by exploring a class of mUltiplicative algorithms. Chapter 1 opens with an introduction to the area of linear design theory. It begins with an outline of a linear regression design problem including properties of the information matrix of the design. The second half of this chapter focuses on several design criteria and their properties. This part consists of two cases: when interest is in inference about all of the parameters of the model and when interest is in some of these parameters. The criteria include D, A, 0, E, D A, L (linear) and EAoptimality. Chapter 2 considers classes of optimisation problems. These include problems [labelled (PI), (P2)] in which the aim is to find an optimising distribution p. In examples of problem (P2) p is seen to define a distribution on a design space. Optimality conditions are determined for such optimisation problems. The emphasis is on a differential calculus approach in contrast to a Lagrangian one. An important tool is the directional derivative F{p, q} of a criterion function ¢(.) at p in the direction of q. The properties of {p, q} are studied, differentiability is expressed in terms of it, and further properties are considered when differentiability is defined. The chapter ends with providing some optimality theorems based on the results of the previous sections. Chapter 3 proposes a class of multiplicative algorithms for these problems. Iterations are of the form· p\r+1) ex p\r)f(x\r») where x\r) = d\r) or F(r) and .} }}' } } j d)r) = a¢/aPj while Ft) = F{p(r),ej} = d)r)  ~p~r)d~r) (a vertex directional ~ derivative) at p = p(r) and f(.) satisfies some suitable properties (positive and strictly increasing) and may depend on one or more free parameters. We refer to this as algorithm (3.1) [the label it is assigned]. These iterations neatly satisfy theconstraints of problems (PI), (P2). Some properties of this algorithm are demonstrated. Chapter 4 focuses on an estimation problem which in the first instance is a seeming generalisation of problem (PI). It is an example of an optimisation problem [labelled (P3) in chapter 2] with respect to variables which should be nonnegative and satisfy several linear constraints. However, it can be transformed to an example of problem (P2). The problem is that of determining maximum likelihood estimates under a hypothesis of marginal homogeneity for data in a square contingency table. The case of a 3 x 3 and of a 4 x 4 contingency table are considered. Chapter 5 investigates the performance of the above algorithm in constructing optimal designs by exploring a variety of choices of f(.) including a class of functions based on a distribution function. These investigations also explore various choices of the argument of f(.). Convergence of the above algorithm are compared for these choices of f(.) and it's argument. Convergence rates can also be controlled through judicious choice of free parameters. The work for this chapter along with the work in chapter 4 has appeared in Mandai and Torsney (2000a). Chapter 6 explores more objective choices of f(.). It mainly considers two approaches  approach I and approach II to improve convergence. In the first f(.) is based on a function h(.) which can have both positive and negative arguments. This approach is appropriate when taking Xj in f(xj) to be Pj , since these vertex directional derivatives being 'centred' on zero, take both positive and negative values. The second bases f(.) on a function g(.) defined only for positive arguments. This is appropriate when taking Xj to be dj if theRe partial derivatives are positive as in the case with design criteria. These enjoy improved convergence rates. Chapter 7 is devoted to a more powerful improvement  a 'clustering approach'. This idea emerges while running algorithm (3.1) in a design space which is a discretisation of a continuous space. It can be observed that 'clusters' start forming in early iterations of the above algorithm. Each cluster centres on a support point of the optimal design on the continuous space. The idea is that, at an appropriate iterate p(r), the single distribution p(r) should be replaced by conditional distributions within clusters and a marginal distribution across the clusters. This approach is formulated for a general regression problem and, then is explored through several regression models, namely, trigonometric, quadratic, cubic, quartic and a secondorder model in two design variables. Improvements in convergence are seen considerably for each of these examples. Chapter 8 deals with the problem of finding an 'approximate' design maximising a criterion under a linear model subject to an equality constraint. The constraint is the equality of variances of the estimates of two linear functions (gT fl. and !l fl.) of the parameters of interest. The criteria considered are D, D A and Aoptimality, where A = [g, QJT. Initially the Lagrangian is formulated but the Lagrange parameter is removed through a substitution, using linear equation theory, in an approach which transforms the constrained optimisation problem to a problem of maximising two functions (Q and G) of the design weights simultaneously. They have a common maximum of zero which is simultaneously attained at the constrained optimal design weights. This means that established algorithms for finding optimising distributions can be considered. The work for this chapter has appeared in Torsney and MandaI (2000). Chapter 9 concludes with a brief review of the main findings of the thesis and a discussion of potential future work on three topics: estimation problems, optimisation with respect to several distributions and constrained optimisation problems.
