A deviance function for the quasi-likelihood method

SUMMARY: We introduce a deviance function that can be used in conjunction with the quasi-likelihood method. The need for such functions arises when the quasi-log likelihood function is not uniquely defined. The deviance is obtained by projecting a pair of centred likelihood ratios onto the direct sum of two Hilbert spaces spanned by the observations. Locally at the null and the alternative hypotheses, the deviance function is equivalent to the quasi-log likelihood ratio provided that the latter is uniquely defined. Like the quasi-log likelihood ratio, it is invariant, antisymmetric and linear in the observations. It can be defined for both independent and dependent observations. In certain situations, when the quasi-score has multiple roots, the confidence set based on the deviance is better than that based on the score test. The deviance also induces a divergence measure between two sets of moments, which resembles Jeffreys divergence between two probability measures.