Strong consistency of least-squares estimators in the monotone regression model with stochastic regressors (Q1094794)

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 \(\theta_ 0\).