Construction of optimising distributions with applications in estimation and optimal design

Mandal, Saumendranath (2000) Construction of optimising distributions with applications in estimation and optimal design. PhD thesis, University of Glasgow.

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Abstract

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 EA-optimality.
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 second-order 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 A-optimality,
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.

Item Type: Thesis (PhD)
Qualification Level: Doctoral
Additional Information: I acknowledge the financial support of a University of Glasgow Scholarship and a UK CVCP Overseas Research Student (ORS) Award (96017016).
Subjects: H Social Sciences > HA Statistics
Q Science > QA Mathematics
Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics
Supervisor's Name: Torsney, Doctor Ben
Date of Award: September 2000
Depositing User: Mrs Marie Cairney
Unique ID: glathesis:2000-7084
Copyright: Copyright of this thesis is held by the author.
Date Deposited: 11 Feb 2016 11:44
Last Modified: 22 Apr 2016 12:16
URI: http://theses.gla.ac.uk/id/eprint/7084

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