Abstract
A composite likelihood is a combination of low-dimensional likelihood objects, and is useful in applications in which the data have a complex structure. The construction of a composite likelihood is crucial, affecting both the computing and the statistical properties of the resulting estimator. Despite this, there is no universal rule for combining low-dimensional likelihood objects that is statistically justified and fast in execution. This study develops a methodology to select and combine the most informative low-dimensional likelihoods from a large set of candidates, while estimating the parameters. The proposed procedure minimizes the distance between composite likelihood and full likelihood scores, subject to a computing cost constraint. The selected composite likelihood is sparse in the sense that it contains relatively few informative sub-likelihoods, and the noisy terms are dropped. The resulting estimator is found to have an asymptotic variance close to that of the minimum-variance estimator constructed using all of the low-dimensional likelihoods.