Small Sample Properties of a Class of Nonlinear Least Squares Problems

Abstract

This paper discusses the least squares fitting of models consisting of a weighted sum of functions, belonging to the same parametric family and non linear in their parameters, to relatively small numbers of observations disturbed by additive errors. The parameters to be estimated are the weights and the nonlinear parameters. The number of terms in the sum is referred to as the order of the model. It is shown that any stationary point of the least squares criterion for a model of a particular order generates stationary points of the least squares criterion for models of the same family of any higher order. Numerical examples show that under influence of the errors such a lower order stationary point may become the absolute and only minimum. A least squares solution does, therefore, not necessarily exist for a particular order, not even for the true one. For the cases considered an explanation of this phenomenon is given.