  Many scientific and engineering applications require fitting regression
models that are nonlinear in the parameters. Advances in computer hardware and
software in recent decades have made it easier to fit such models. Relative to
fitting regression models that are linear in the parameters, however, fitting
nonlinear regression models is more complicated. In particular, software like
the $\texttt{nls}$ R function requires care in how the model is parameterized
and how initial values are chosen for the maximum likelihood iterations. Often
special diagnostics are needed to detect and suggest approaches for dealing
with identifiability problems that can arise with such model fitting. When
using Bayesian inference, there is the added complication of having to specify
(often noninformative or weakly informative) prior distributions. Generally,
the details for these tasks must be determined for each new nonlinear
regression model. This paper provides a step-by-step procedure for specifying
these details for any appropriate nonlinear regression model. Following the
procedure will result in a numerically robust algorithm for fitting the
nonlinear regression model. We illustrate the methods with three different
nonlinear models that are used in the analysis of experimental fatigue data and
we include two detailed numerical examples.
