Some problems in estimation in mixed linear models

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. iv 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. Related publications Several papers based on the results of this thesis have already been published or prepared for publication. These are Richardson, A.M. (1994). Bounded influence estimation in the mixed linear model. Submitted. Richardson, A.M. (1995). A comparison of robust likelihood methods for calculating variance components. Unpublished manuscript. Richardson, A.M. and Welsh, A.H. (1994a). Asymptotic properties of restricted maximum likelihood (REML) estimates for hierarchical mixed linear models. Australian Journal o f Statistics 36 31-43. Richardson, A.M. and Welsh, A.H. (1994b). Covariate screening in mixed linear models. Submitted. Richardson, A.M. and Welsh, A.H. (1995). Robust restricted maximum likelihood in mixed linear models. Biometrics to appear.