Nonlinear Least Squares

Abstract

In this chapter we extend our methods based on the forward search to regression models that are nonlinear in the parameters. Estimation is still by least squares although now iterative methods have to be used to find the parameter values minimizing the residual sum of squares. Even with normally distributed errors, the parameter estimates are not exactly normally distributed and contours of the sum of squares surfaces are not exactly ellipsoidal. The consequent inferential problems are usually solved by linearization of the model by Taylor series expansion, in effect ignoring the nonlinear aspects of the problem. The next section gives an outline of this material, booklength treatments of which are given by Bates and Watts (1988) and by Seber and Wild (1989). Both books describe the use of curvature measures to assess the effect of nonlinearity on approximate inferences using the linearized model. Since we find it informative to monitor measures of curvature during the forward search, we present a summary of the theory in §5.1.2. Ratkowsky (1983) uses measures of curvature to find parameter transformations that reduce curvature and so improve the performance of nonlinear least squares fitting routines.