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Biometrika Advance Access published online on August 5, 2007
Biometrika, doi:10.1093/biomet/asm040
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Copyright © 2007 Biometrika Trust

Article

Integrated likelihood functions for non-Bayesian inference

Thomas A. Severini

Department of Statistics, Northwestern University, Evanston, Illinois 60208-4070, U.S.A.

severini{at}northwestern.edu

Received for publication 1 November 2005. Revision received 1 November 2006.
  Abstract

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.

Key Words: Modified profile likelihood • Nuisance parameter • Orthogonal parameters • Reference prior


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