A Note on Information Bias and Efficiency of Composite Likelihood

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