Clarke, Brenton Ross

### Description

This thesis is concerned with the asymptotic theory of general
M-estimators and some minimal distance estimators. Particular attention
is paid to uniform convergence theory which is used to prove limit
theorems for statistics that are usually implicitly defined as solutions
of estimating equations.
The thesis is divided into eight chapters and into three main
sections. In Section A the theory of convergence is studied as a prelude
to validating the use of the particular M-estimators...[Show more] given in Section B
and C. Section B initially covers the view of robustness of Hampel
(1968) but places more emphasis on the application of the notions of
differentiability of functionals and on M-estimators of a general parameter
that are robust against "tail" contamination. Sections A and B
establish a base for a comparison of robustness and application aspects
of minimal distance estimators, particularly with regard to their
application to estimating mixtures of normal distributions. An important
application of this is illustrated for the analysis of seismic data.
This constitutes Section C.
Chapter 1 is devoted to the study of uniform convergence theorems
over classes of functions and sets allowing also the possibility that
the underlying probability mechanism may be from a specified family.
A new Glivenko-Cantelli type theorem is proved which has applications
later to weakening differentiability requirements for the convergence
of loss functions used in this thesis.
For implicitly defined estimators it is important to clearly
identify the estimator. By uniform convergence, asymptotic uniqueness
in regions of the parameter space of solutions to estimating equations
can be established. This then justifies the selection of solutions through appropriate statistics, thus defining estimators uniquely for
all samples. This comes under the discussion of existence and consistency
in Chapter 2. Chapter 3 includes central limit theorems and
the law of the iterated logarithm for the general M-estimator,
established under various conditions, both on the loss function and on
the underlying distribution. Uniform convergence plays a central role
in showing the validity of approximating expansions. Results are shown
for both univariate and multivariate parameters. Arguments for the
univariate parameter are often simpler or require weaker conditions.
Our study of robustness is both of a theoretical and quantitative
nature. Weak continuity and also Frechet differentiability with respect
to Prokhorov, Levy and Kolmogorov distance functions are established
for multivariate M-functionals under similar but necessarily stronger
conditions than those required for asymptotic normality. Relationships
between the conditions imposed on the class of loss functions in order
to attain Frechet differentiability and those necessary and sufficient
conditions placed on classes of functions for which uniform convergence
of measures hold can be shown. Much weaker conditions exist for almost
sure uniform convergence and this goes part way to explaining the
restrictive nature of this functional derivative approach to showing
asymptotic normality.
In Chapter 5 the notion of a set of null influence is emphasized.
This can be used to construct M-functionals robust (in terms of asymptotic
bias and variance) against contamination in the "tails" of a distribution.
This set can depend on the parameter being estimated and in this sense
the resulting estimator is adaptive. Its construction is illustrated
in Chapter 6 for the estimation of scale. Robustness against "tail"
contamination is illustrated by numerical comparison with other M-estimators. Particular applications are given to inference in the
joint estimation of location and scale where it is important to identify
the root to the M-estimating equations. Techniques justified by uniform
convergence are used here. Uniform convergence also lends itself to the
use of a graphical method of plotting "expectation curves". It can be
used for either identifying the M-estimator from multiple solutions of
the defining equations or in large samples (e.g. > 50) as a visual indication
of whether the fitted model is a good approximation for the underlying
mechanism. Theorems based on uniform convergence are given that
show a domain of convergence (numerical analysis interpretation) for the
Newton-Raphson iteration method applied to M-estimating equations for
the location parameter when redescending loss functions are used.
The M-estimator theory provides a common framework whereby some
minimal distance methods can be compared. Two established L₂ minimal
distance estimators are shown to be general M-estimators. In particular
a Cramer-Von Mises type distance estimator is shown to be qualitatively
robust and have good small sample properties. Its applicability to
some new mixture data from geological recordings, which clearly requires
robust methods of analysis is demonstrated in Chapters 7 and 8.

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