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Quasi-likelihood and optimal estimation. (English) Zbl 0671.62007
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)].
Reviewer: K.I.Yoshihara

##### MSC:
 62A01 Foundations and philosophical topics in statistics
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