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A composite likelihood is a combination of low-dimensional likelihood objectsuseful in applications where the data have complex structure. Althoughcomposite likelihood construction is a crucial aspect influencing bothcomputing and statistical properties of the resulting estimator, currentlythere does not seem to exist a universal rule to combine low-dimensionallikelihood objects that is statistically justified and fast in execution. Thispaper develops a methodology to select and combine the most informativelow-dimensional likelihoods from a large set of candidates while carrying outparameter estimation. The new procedure minimizes the distance betweencomposite likelihood and full likelihood scores subject to a constraintrepresenting the afforded computing cost. The selected composite likelihood issparse in the sense that it contains a relatively small number of informativesub-likelihoods while the noisy terms are dropped. The resulting estimator isfound to have asymptotic variance close to that of the minimum-varianceestimator constructed using all the low-dimensional likelihoods.

Fast construction of optimal composite likelihoods