Robustness properties of marginal composite likelihood estimators

Composite likelihoods are a class of alternatives to the full likelihood which are widely used in many situations in which the likelihood itself is intractable. A composite likelihood may be computed without the need to specify the full distribution of the response, which means that in some situations the resulting estimator will be more robust to model misspecification than the maximum likelihood estimator. The purpose of this note is to show that such increased robustness is not guaranteed. An example is given in which various marginal composite likelihood estimators are inconsistent under model misspecification, even though the maximum likelihood estimator is consistent.