Let \(\Theta\) be an open subset of \(R^ p\) and let \({\mathcal P}=\{{\mathcal P}_{\theta}\}\) be a union of parametric families of probability measures, each family being indexed by the same parameter \(\theta\in \Theta\). Let \(\{X_ t:\) \(0\leq t\leq T\}\) be a sample in discrete or continuous time which is drawn from some process taking values in \(R^ r\) and with \({\mathcal P}_{\theta}\in {\mathcal P}\). Further, let \({\mathcal G}\) be the class of zero mean, square integrable estimating functions \(G_ T=G_ T(\{X_ t:\) \(0\leq t\leq T\},\theta)\) for which E \(G_ T(\theta)=0\) for each \({\mathcal P}_{\theta}.\)
The authors consider three equivalent properties of an estimating function belonging to a subclass of \({\mathcal G}\) and then by referring to these properties define the optimality of estimating functions, the quasi-score function, the quasi-likelihood equation and a maximum quasi- likelihood estimator when sample sizes are fixed. Next, they define “optimal in the asymptotic sense” and consider a criterion to be so within a subclass of \({\mathcal G}\) consisting of martingale estimating functions which are square integrable martingales for each \({\mathcal P}_{\theta}\). They also discuss applications to stochastic processes.
The results extend those of the first author [Biometrika 72, 419-428 (1985; Zbl 0584.62135)] and J. E. Hutton and P. I. Nelson [Stochastic Processes Appl. 22, 245-257 (1986; Zbl 0616.62113)].