Parameter Estimation in Semi-Linear Models Using a Maximal Invariant Likelihood Function

In this paper, we consider the problem of estimation of semi-linear regression models. Using invariance arguments, Bhowmik and King (2001) have derived the probability density functions of the maximal invariant statistic for the nonlinear component of these models. Using these density functions as likelihood functions allows us to estimate these models 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. We report the results of a simulation study conducted to compare the accuracy of this approach with full maximum likelihood estimation. We find maximising the maximal invariant likelihood function typically results in less biased and lower variance estimates than those from full maximum likelihood.