A Note on Extended Quasi-Likelihood

We study the method of extended quasi-likelihood estimation and inference of a variance function recently proposed by Nelder Pregibon 1987. The estimates are inconsistent in general, and the test levels can be biased, but in many cases such as the exponential family the inconsistency and bias will not be a major concern. Extended quasi-likelihood is compared with Carroll Rupperts 1982 pseudo-likelihood method, which gives consistent estimates and, when slightly modified, asymptotically unbiased tests. We quantify the showing in this instance that the two estimates are closely related and may be asymptotically equivalent in many important cases. However, in some cases outside the exponential family, an asymptotic bias can persist. Heteroscedastic regression model.