Generalized Quasi-Likelihood (GQL) Inferences

QL Estimation for Independent Data

For i = 1, …, K, let Y _i denote the response variable for the ith individual, and x _i = (x _i1, …, x _iv , …, x _ip )′ be the associated p − dimensional covariate vector. Also, let β be the p − dimensional vector of regression effects of x _i on y _i . Further suppose that the responses are collected from K independent individuals. It is understandable that if the probability distribution of Y _i is not known, then one can not use the well known likelihood approach to estimate the underlying regression parameter β. Next suppose that only two moments of the data, that is, the mean and the variance functions of the response variable Y _i for all i = 1, …, K, are known, and for a known functional form a( ⋅), these moments are given by

$$E[{Y }_{i}] = a'({\theta }_{i})\;\mbox{ and}\;\mbox{ var}[{Y }_{i}] = a''({\theta }_{i}),$$

(1)

where for a link function h( ⋅), θ _i = hx′ _i β, and a′(θ _i ) and a′(θ _i ) are the first and second order derivatives of a(θ...