Biometrika Advance Access originally published online on August 5, 2007
Biometrika 2007 94(3):529-542; doi:10.1093/biomet/asm040
Copyright © 2007 Biometrika Trust
Articles |
Integrated likelihood functions for non-Bayesian inference
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.
Consider a model with parameter 
= (
, 
), where is the parameter of interest, and let L(, ) denote the likelihood function. One approach to likelihood inference for is to use an integrated likelihood function, in which is eliminated from L(, ) by integrating with respect to a density function 
(|). The goal of this paper is to consider the problem of selecting (|) 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 (|) should be chosen by finding a nuisance parameter 
that is unrelated to and then taking the prior density for to be independent of . 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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