  Does the asymptotic variance of the maximum composite likelihood estimator of
a parameter of interest always decrease when the nuisance parameters are known?
Will a composite likelihood necessarily become more efficient by incorporating
addi- tional independent component likelihoods, or by using component
likelihoods with higher dimension? In this note we show through illustrative
examples that the an- swer to both questions is no, and indeed the opposite
direction might be observed. The role of information bias is highlighted to
understand the occurrence of these paradoxical phenomenon.
