Parameter estimation in linear models with a nonlinear component using a maximal invariant likelihood function

In this paper, we consider the problem of estimation of a regression model with both linear and nonlinear components. Using invariance arguments, Bhowmik and King (2001) have derived the probability density function of the maximal invariant statistic for the nonlinear component of this model. Clearly this density function can be used as a likelihood function for the nonlinear component. This allows us to estimate the model in a two step process. First the nonlinear component parameters are estimated by maximising the maximal invariant likelihood function. Then the nonlinear component, with the parameter values replaced by estimates, is treated as a regressor and ordinary least squares is used to estimate the remaining parameters. Alternatively the full likelihood of the complete model can be maximised to obtain the standard maximum likelihood estimates. We report the results of a simulation study conducted to compare the accuracy of these two approaches