Abstract: Although the properties of inferences based on a composite likelihood are well established, they can be surprising, leading to misleading results. In this note, we show by example that the variance of a maximum composite likelihood estimator can increase when the nuisance parameters are known, rather than estimated. In addition, we show that estimators based on more independent component likelihoods can be less efficient than those based on fewer such likelihoods, and that incorporating higher-dimensional marginal densities can also lead to a less efficient inference. The role of information bias is highlighted to understand why these paradoxical phenomena occur.

Key words and phrases: Bartlett's second identity, estimating function, godambe information matrix, nuisance parameter, pairwise likelihood.