Integrated likelihood functions for non-Bayesian inference

Consider a model with parameter theta = (psi, lambda), where psi is the parameter of interest, and let L(psi, lambda) denote the likelihood function. One approach to likelihood inference for psi is to use an integrated likelihood function, in which lambda is eliminated from L(psi, lambda) by integrating with respect to a density function pi(lambda|psi). The goal of this paper is to consider the problem of selecting pi(lambda|psi) so that the resulting integrated likelihood function is useful for non- Bayesian likelihood inference. The desirable properties of an integrated likelihood function are analyzed and these suggest that pi(lambda|psi) should be chosen by finding a nuisance parameter phi that is unrelated to psi and then taking the prior density for phi to be independent of psi. Such an unrelated parameter is constructed and the resulting integrated likelihood is shown to be closely related to the modified profile likelihood.