# Some problems in estimation in mixed linear models

## Date

1995

## Authors

Richardson, Alice

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## Abstract

This thesis is concerned with the properties of classical estimators of the
parameters in mixed linear models, the development of robust estimators, and the
properties and uses of these estimators. The first chapter contains a review of
estimation in mixed linear models, and a description of four data sets that are used to
illustrate the methods discussed.
In the second chapter, some results about the asymptotic distribution of the
restricted maximum likelihood (REML) estimator of variance components are stated
and proven. Some asymptotic results are also stated and proven for the associated
weighted least squares estimtor of fixed effects. Central limit theorems are obtained
using elementary arguments with only mild conditions on the covariates in the fixed
part of the model and without having to assume that the data are either normally or
spherically symmetrically distributed. It is also shown that the REML and maximum
likelihood estimators of variance components are asymptotically equivalent.
Robust estimators are proposed in the third chapter. These estimators are M -
estimators constructed by applying weight functions to the log-likelihood, the
restricted log-likelihood or the associated estimating equations. These functions
reduce the influence of outlying observations on the parameter estimates. Other
suggestions for robust estimators are also discussed, including Fellner's method. It is
shown that Fellner's method is a direct robustification of the REML estimating
equations, as well as being a robust version of Harville's algorithm, which in turn is
equivalent to the expectation-maximisation (EM) algorithm of Dempster, Laird and
Rubin.
The robust estimators are then modified in the fourth chapter to define bounded
influence estimators, also known as generalised M or GM estimators in the linear
regression model. Outlying values of both the dependent variable and continuous
independent variables are downweighted, creating estimators which are analogous to
the GM estimators of Mallows and Schweppe. Some general results on the asymptotic
properties of bounded influence estimators (of which maximum likelihood, REML and
the robust methods of Chapter 3 are all special cases) are stated and proven. The
method of proof is similar to that employed for the classical estimators in Chapter 2.
Chapter 5 is concerned with the practical problem of selecting covariates in
mixed linear models. In particular, a change of deviance statistic is proposed which
provides an alternative to likelihood ratio test methodology and which can be applied in
situations where the components of variance are estimated by REML. The deviance is
specified by the procedure used to estimate the fixed effects and the estimated
covariance matrix is held fixed across different models for the fixed effects. The
distribution of the change of deviance is derived, and a robustification of the change of
deviance is given. Finally, in Chapter 6 a simulation study is undertaken to investigate the
asymptotic properties of the proposed estimators in samples of moderate size. The
empirical influence function of some of the estimators is studied, as is the distribution
of the change of deviance statistic. Issues surrounding bounded influence estimation
when there are outliers in the independent variables are also discussed.

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Thesis (PhD)

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