Nonlinear Regression Models

Abstract

In this chapter we shall consider the problem of estimating the parameters of a nonlinear regression model with additive disturbance term under a number of alternative assumptions about the distributions of errors and regressors. In section 3.1 we first discuss the least squares estimation theory of Jennrich (1969), but in a slightly more general setting than Jennrich did. Then, at the end of this section, we shall weaken some assumptions and especially we show what happens if the error distribution is symmetric stable with characteristic exponent α < 2. In section 3.2 a nonlinear robust two-stage M-estimation method is presented. This method is especially appropriate when the second moment of the error distribution is infinite. But even when the usual assumptions about the error distribution are satisfied, but the error distribution is leptocurtic, which means that the fourth moment of this error distribution is larger than three times the square of the second moment, then robust M-estimation turns out to be asymptotically more efficient than least squares. Both least squares estimation and robust M-estimation requires some assumptions about the distributions of the regressors. In section 3.3 we therefore introduce a weighted robust two-stage M-estimation method which does not need any assumption about finiteness of moments of errors and regressors. The first-stage estimators of both robust M-estimation methods are random vector-functions of a scaling parameter. In section 3.4.1 it is shown that these vector-functions are (pseudo) uniformly consistent on a compact set of scaling parameters.