Some years ago one of the authors (G.A.F.S.) asked a number of applied
statisticians how they got on with fitting nonlinear models. The answers were
generally depressing. In many cases the available computing algorithms for
estimation had unsatisfactory convergence properties, sometimes not converging
at all, and there was some uncertainty about the validity of the linear
approximation used for inference. Furthermore, parameter estimates sometimes
had undesirable properties. Fortunately the situation has improved over recent
years because of two major developments. Firstly, a number of powerful
algorithms for fitting models have appeared. These have been designed to handle
"difficult" models and to allow for the various contingencies that can arise in
iterative optimization. Secondly, there has been a new appreciation of the role of
curvature in nonlinear modeling and its effect on inferential procedures.
Curvature comes in a two-piece suite: intrinsic curvature, which relates to the
geometry of the nonlinear model, and parameter-effects curvature, which
depends on the parametrization of the model. The effects of these curvatures have
recently been studied in relation to inference and experimental design. Apart from
a couple of earlier papers, all the published literature on the subject has appeared
since 1980, and it continues to grow steadily. It has also been recognized that the
curvatures can be regarded as quantities called connection coefficients which
arise in differential geometry, the latter providing a unifying framework for the
study of curvature. Although we have not pursued these abstract concepts in
great detail, we hope that our book will at least provide an introduction and make
the literature, which we have found difficult, more accessible.