The class of composite likelihood functions provides a flexible and powerful toolkit to
carry out approximate inference for complex statistical models when the full likelihood
is either impossible to specify or unfeasible to compute. However, the strenght of the
composite likelihood approach is dimmed when considering hypothesis testing about a
multidimensional parameter because the finite sample behavior of likelihood ratio, Wald,
and score-type test statistics is tied to the Godambe information matrix. Consequently
inaccurate estimates of the Godambe information translate in inaccurate p-values. In this
paper it is shown how accurate inference can be obtained by using a fully nonparametric
saddlepoint test statistic derived from the composite score functions. The proposed statis-
tic is asymptotically chi-square distributed up to a relative error of second order and does
not depend on the Godambe information. The validity of the method is demonstrated
through simulation studies.
