Resumen
In this paper, we consider the problem of estimation of semi-linear regression models. Using invariance arguments, Bhowmik and King [2007. Maximal invariant likelihood based testing of semi-linear models. Statist. Papers 48, 357–383] derived the probability density function of the maximal invariant statistic for the non-linear component of these models. Using this density function as a likelihood function allows us to estimate these models in a two-step process. First the non-linear component parameters are estimated by maximising the maximal invariant likelihood function. Then the non-linear 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 and maximum profile-marginal 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.

Resumen
This paper deals with testing for non-linearity in a regression model with one possibly non-linear component being estimated non-parametrically using smoothing splines. We propose two new variance–covariance based tests for detecting non-linearity applying a likelihood ratio hypothesis testing approach. The first test is for the inclusion of a possibly non-linear component and the second one is for linearity of a possibly non-linear component. The tests are based on a stochastic model in state space form given by Wahba (J. Roy. Statist. Soc. Ser. B 40 (1978) 364), Wecker and Ansley (J. Amer. Statist. Assoc. 78 (1983) 81) and de Jong and Mazzi (Modeling and smoothing unequally spaced sequence data, University of York and University of British Columbia, Unpublished paper) for which smoothing splines provide an optimal estimate. Pitrun (A smoothing spline approach to non-linear interface for time series, Department of Econometrics and Business Statistics, Monash University, Unpublished Ph.D. thesis) derived the variance–covariance structure of this model, which allows the use of a marginal likelihood approach. This leads naturally to marginal-likelihood based likelihood ratio tests for non-linearity. Small sample properties of the new tests have been investigated via Monte Carlo studies.