On divergence tests for composite hypotheses under composite likelihood

It is well-known that in some situations it is not easy to compute the likelihood function as the datasets might be large or the model is too complex. In that contexts composite likelihood, derived by multiplying the likelihoods of subjects of the variables, may be useful. The extension of the classical likelihood ratio test statistics to the framework of composite likelihoods is used as a procedure to solve the problem of testing in the context of composite likelihood. In this paper we introduce and study a new family of test statistics for composite likelihood: Composite ϕ$$\phi $$ -divergence test statistics for solving the problem of testing a simple null hypothesis or a composite null hypothesis. To do that we introduce and study the asymptotic distribution of the restricted maximum composite likelihood estimate.