Integrated likelihood inference in semiparametric regression models

Consider a linear semiparametric regression model with normal errors in which the mean function depends on two parameters, a p-dimensional regression parameter, which is the parameter of interest, and an unknown function, which is a nuisance parameter. We consider estimation of the parameter of interest using an integrated likelihood function, in which the nuisance parameter is eliminated from the likelihood function by averaging with respect to some distribution. Here we take this distribution to be a Gaussian process with a given covariance function, whichmay depend on additional parameters. Likelihood inference based on the resulting integrated likelihood is considered and the properties of the score statistic based on the integrated likelihood, the maximum integrated likelihood estimator, and the integrated likelihood ratio statistic are presented. The methodology is illustrated on two examples.