Integrated likelihood functions for non-Bayesian inference

Abstract

Consider a model with parameter θ = (ψ, λ), where ψ is the parameter of interest, and let L (ψ, λ) denote the likelihood function. One approach to likelihood inference for ψ is to use an integrated likelihood function, in which λ is eliminated from L (ψ, λ) by integrating with respect to a density function π(λ|ψ). The goal of this paper is to consider the problem of selecting π(λ|ψ) 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 π(λ|ψ) should be chosen by finding a nuisance parameter ϕ that is unrelated to ψ and then taking the prior density for ϕ to be independent of ψ. Such an unrelated parameter is constructed and the resulting integrated likelihood is shown to be closely related to the modified profile likelihood.