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Strong consistency of least-squares estimators in the monotone regression model with stochastic regressors. (English) Zbl 0631.62076

Let \(y_ t=f(x_ t,\theta_ 0)+\epsilon_ t\), \(t=1,2,..\). be a nonlinear regression model, where \(\epsilon_ t\), \(t=1,2,..\). form a martingale difference sequence; \(f(x,\theta)=\theta (x)\) with \(\theta\) : \(R^ d\to R\) ranging in the class of all measurable, monotone increasing functions on \({\mathcal X}\subseteq R^ d\) \((x_ i\leq y_ i\), \(i=1,...,d\) implies \(\theta\) (x)\(\leq \theta (y)).\)
Under some conditions, every sequence \({\hat \theta}_ T\), \(T=1,2,..\). of monotone least-square estimators is strongly consistent on int(\({\mathcal X})\), i.e. with probability one, \({\hat \theta}_ T(x)\to \theta_ 0(x)\) as \(T\to \infty\) for all continuity points \(x\in int({\mathcal X})\) of \(\theta_ 0\).
Reviewer: N.Leonenko

MSC:

62J02 General nonlinear regression
62F12 Asymptotic properties of parametric estimators
60G42 Martingales with discrete parameter
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