Asymptotic Theory of Estimation in Nonlinear Regression

Abstract

Nonlinear regression models occur frequently in the modeling of stochastic phenomena. Several examples of such modeling are given by Bard (1974). The study of asymptotic properties of the least squares estimator (LSE) for parameters occurring in nonlinear regression has been the subject of investigation by several authors in view of the fact that it is, in general, difficult to obtain the exact distribution of the LSE for any fixed sample. Malinvaud (1970), Jennrich (1969), Bunke and Schmidt (1980) and Wu (1981) are a few authors among the many who have investigated the asymptotic properties of the LSE in nonlinear regression. Schmidt (1982) has given a survey of testing of hypotheses in nonlinear regression. All the earlier work cited above on asymptotic distribution theory for least squares estimators in nonlinear regression models assume regularity conditions which include, in particular, the condition on the twice differentiability of the regression function with respect to the parameter in a neighbourhood of the true value in addition to other conditions. Schmidt (1982, p. 18) says that up to my knowledge, there is no idea how to prove the asymptotic normality of the LSE when g(x,θ) is not differentiable with respect to θ since all the proofs use the normal equations. Recently we have given an alternate approach for the study of asymptotic distribution theory. The least squares is considered as a stochastic process in the parameter and the limiting distribution, if any, is obtained via the study of weak convergence of the least squares process.