A CAVEAT ON THE ROBUSTNESS OF COMPOSITE LIKELIHOOD ESTIMATORS: THE CASE OF A MIS-SPECIFIED RANDOM EFFECT DISTRIBUTION

Composite likelihoods are a class of alternatives to the full likelihood which may be used for inference in many situations where the likelihood itself is intractable. A composite likelihood estimator will be robust to certain types of model misspecification, since it may be computed without the need to specify the full distribution of the response. This potential for increased robustness has been widely discussed in recent years, and is considered a secondary motivation for the use of composite likelihood. The purpose of this paper is to show that there are some situations in which a composite likelihood estimator may actually suffer a loss of robustness compared to the maximum likelihood estimator. We demonstrate this in the case of a generalized linear mixed model under misspecification of the random effect distribution. As the amount of information available on each random effect increases, we show that the maximum likelihood estimator remains consistent under such misspecification, but various marginal composite likelihood estimators are inconsistent. We conclude that composite likelihood estimators cannot in general be claimed to be more robust than the maximum likelihood estimator.