Adjusting composite likelihood ratio statistics

Composite likelihood may be useful for approximating likelihood based inference when the full likelihood is too complex to deal with. Stemming from a misspecified model, inference based on composite likelihood requires suitable corrections. Here we focus on the composite likelihood ratio statistic for a multidimensional parameter of interest, and we propose a parameterization invariant adjustment that allows reference to the usual asymptotic chi-square distribution. Two examples dealing with pairwise likelihood are analysed through simulation.