Note on information bias and efficiency of composite likelihood

TL;DR

This paper investigates the efficiency and bias of composite likelihood estimators, revealing that more information or higher-dimensional components do not always lead to increased efficiency, due to information bias effects.

Contribution

It demonstrates through examples that composite likelihood efficiency can decrease with additional components or known nuisance parameters, highlighting the role of information bias.

Findings

Adding independent component likelihoods does not always improve efficiency

Knowing nuisance parameters does not necessarily increase estimator precision

Information bias can cause paradoxical efficiency outcomes

Abstract

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.