Doctoral thesis
Open access

High breakdown inference for mixed linear models

ContributorsCopt, Samuel
Defense date2004-06

Mixed linear models are used to analyze data in many settings. These models have in most cases a multivariate normal formulation. The maximum likelihood estimator (MLE) or the residual MLE (REML) are usually chosen to estimate the parameters. However, the latter are based on the strong assumption of exact multivariate normality. Welsh and Richardson (1997) have shown that these estimators are not robust to small deviations from the multivariate normality. This means, in practice, that a small proportion of data (even only one) can drive the value of the estimates on their own. We present some of the most used models in the analysis of variance. We introduce the mixed linear model formulation and see that in most cases it is possible to extract independent subvectors of observation. The structure of the covariance matrix is derived for a great variety of models. Since the model is multivariate, we propose in this thesis a high breakdown multivariate robust estimator for very general mixed linear models, that include, for example, covariates. This robust estimator belongs to the class of S-estimators (Rousseeuw and Yohai 1984) from which we can derive the asymptotic properties for inference. We also use it as a diagnostic tool to detect outlying subjects. We derive the estimating equation defining the high breakdown estimator and we describe how it can be computed via a simple iterative algorithm. We study the behavior of the robust estimator through an extensive simulation study. It is compared to the maximum likelihood estimator under a great variety of configuration implying different models, different contamination patterns and different samples size. We also discuss the advantages of this estimator and illustrate its performance with the analysis of four datasets. We also consider robust inference for multivariate hypotheses as an alternative to the classical F-test by using a robust score type test statistic proposed by Heritier and Ronchetti (1994) and study its properties by means of simulations and real data analysis.

Citation (ISO format)
COPT, Samuel. High breakdown inference for mixed linear models. 2004. doi: 10.13097/archive-ouverte/unige:12053
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