An extension of quasi-likelihood estimation

Abstract

Godambe (1985, 1987) obtained the optimal combination of 'orthogonal' estimating functions. Using these results here we extend the concept and technique of quasi-likelihood estimation (Wedderburn, 1974), incorporating possible knowledge of the skewness, kurtosis and higher moments of the underlying distribution. This is done by defining the extended quasi-score function. The definition includes as a special case the quasi-score function, which was implicit in Wedderburn's (1974) quasi-likelihood function but was made explicit as a 'pseudo-score function' by Godambe (1985). A close parallel, conceptual and operational, between the 'extended quasi-score function' and the 'score function' is established. It is pointed out that the former is the natural substitute for the latter when the likelihood function is undefined, as for instance in semi-parametric models.