Issues and Strategies in the Selection of Composite Likelihoods

The composite likelihood method has been proposed and systematically discussed by Besag (1974), Lindsay (1988), and Cox and Reid (2004). This method has received increasing interest in both theoretical and applied aspects. Compared to the traditional likelihood method, the composite likelihood method may be less statistically efficient, but it can be designed so as to be significantly faster to compute and it can be more robust to model misspecification. Although there are a number of ways to formulate a composite likelihood to balance the trade-off between the efficiency and computational price, there does not seem to exist a universal rule for constructing a combination of composite likelihoods that is both computationally convenient and statistically appealing. In this article we present some thoughts on the composite likelihood, drawing on basic knowledge about likelihood and estimating functions. A new efficiency result based on the Hoeffding decomposition of U -statistics is given. A recommendation is given to consider the construction of surrogate density functions as a way to better bridge the gap between likelihood methods and composite likelihood methods.