Quasi-likelihood

• Definition
  Suppose we have independent observations $z_i (i = 1,...,n)$ with expectations $\mu_i$ and variances $V(\mu_i)$, where $V$ is some known function. Later we shall relax this specification and say $var(z_i) \propto V(\mu_i)$. We suppose that for each observation $\mu_i$ is some known function of a set of parameters $\beta_1, \cdots, \beta_r$. Then for each observation we define the quasi-likelihood function $K(z_i, \mu_i)$ by the relation $$ \frac{\partial K(z_i, \mu_i)}{\partial \mu_i} = \frac{z_i - \mu_i}{V(\mu_i)}, $$ or equivalently $$ K(z_i, \mu_i) = \int^{\mu_i} \frac{z_i - \mu_i^`}{V(\mu_i^`)} d\mu_i^` + \text{ function of } z_i. $$