Often, when a data-generating process is too complex to specify fully, a standard likelihood-based inference is not available. However, a composite likelihood can provide an inference based on a partial specification of a data-generating process. Furthermore, its robustness to model specification and computational simplicity makes the composite likelihood method widely applicable. This study conducts a theoretical investigation of the composite likelihood ratio test (CLRT) when the parameters of interest may lie on the boundary of the parameter space. Our main result shows that the limiting distribution of the CLRT is equivalent to that of the likelihood ratio test of a normal mean problem, in which the restricted mean of a multivariate normal distribution is tested based on one observation from a multivariate normal distribution with an inverse Godambe information matrix. Furthermore, we illustrate our general theoretical result by applying it to a variety of examples. Lastly, our simulation results confirm that the limiting distribution of the CLRT performs well in finite samples.
