Modified Likelihood and Related Methods for Handling Nuisance Parameters in the Linear Regression Model

In this paper, different approaches to dealing with nuisance parameters in likelihood based inference are presented and illustrated by reference to the linear regression model with nonspherical errors. The estimator of the error variance using each of the approaches is also derived for the linear regression model with spherical errors. We observe that many of these estimators are unbiased. A theoretical comparison of the likelihood functions is reported and we note that some of them are equivalent. Empirical evidence in the literature indicates that estimators based on the conditional profile likelihood and tests based on the marginal likelihood have better small sample properties compared to those based on other likelihood and message length functions.