Embedded Models in Non-Linear Regression

A problem that is often encountered in non-linear regression is instability in the computation of parameter estimates. Such instability may arise from models with too many parameters, resulting in estimating equations that are ill conditioned and consequent problems of identification. This paper shows that in some cases these difficulties can be explained by the presence of an embedded model, a special case of the original model involving fewer parameters but one that is not readily identified from the parameterization of the model. The paper shows how an embedded model can be identified by making a suitable reparameterization. Moreover this reparameterization removes the problems of numerical instability where an embedded model turns out to be the best fit to the data. The paper suggests how the reparameterization can be chosen systematically. Numerical examples are included to illustrate these points.