Asymptotic Theory of Nonlinear Least Squares Estimation

Abstract

For a linear regression model, the necessary and sufficient condition for the asymptotic consistency of the least squares estimator is known. An analogous condition for the nonlinear model is considered in this paper. The condition is proved to be necessary for the existence of any weakly consistent estimator, including the least squares estimator. It is also sufficient for the strong consistency of the nonlinear least squares estimator if the parameter space is finite. For an arbitrary compact parameter space, its sufficiency for strong consistency is proved under additional conditions in a sense weaker than previously assumed. The proof involves a novel use of the strong law of large numbers in $C(S)$. Asymptotic normality is also established.