A Deviance Function for the Quasi-Likelihood Method
提出一种偏差函数，用于准似然方法中准对数似然函数不唯一的情况；该函数在零假设和备择假设下等价于准对数似然比，且能构造更好的置信集。
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.