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Consistency of M estimates for separable nonlinear regression models

Consider a nonlinear regression model : y_{i}=g(x_{i},θ)+e_{i}, i=1,...,n, where the x_{i} are random predictors x_{i} and θ is the unknown parameter vector ranging in a set Θ\subsetR^{p}. All known results on the consistency of the least squares estimator and in general of M estimators assume that either Θ is compact or g is bounded, which excludes frequently employed models such as the Michaelis-Menten, logistic growth and exponential decay models. In this article we deal with the so-called separable models, where p=p_{1}+p_{2}, θ=(α,β) with α\inA\subsetR^{p_{1}}, β\inB\subsetR^{p_{2},}and g has the form g(x,θ)=β^{T}h(x,α) where h is a function with values in R^{p_{2}}. We prove the strong consistency of M estimators under very general assumptions, assuming that h is a bounded function of α, which includes the three models mentioned above. Key words and phrases: Nonlinear regression, separable models, consistency, robust estimation.