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Simulation and the asymptotics of optimization estimators. (English) Zbl 0698.62031
Consider a model in which the true value \(\theta_ 0\) is implicitly defined as the unique solution to an equation \(G(\theta)=0\) for a suitable vector-valued function. Let \(\{G_ n\}\) be a sequence of not necessarily continuous random criterion functions that converge to G in some sense and for each n let \({\hat \theta}{}_ n\) be an estimator of \(\theta_ 0\) that makes \(G_ n({\hat \theta}_ n)\) as close to zero as possible.
The authors prove a consistency theorem and a central limit theorem for \({\hat \theta}{}_ n\) under uniformity conditions. To check these conditions, the authors introduce the concept of “Euclidean for an envelope”. The method used here is particularly useful for applications such as the study of simulation estimators where the criterion function can have discontinuities. Two examples are also shown.
Reviewer: K.-i.Yoshihara

62F12 Asymptotic properties of parametric estimators
60F05 Central limit and other weak theorems
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