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Biometrika 1998 85(3):507-522; doi:10.1093/biomet/85.3.507
© 1998 by Biometrika Trust
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Likelihood functions for inference in the presence of a nuisance parameter

THOMAS A. SEVERINI

Department of Statistics, Northwestern University Evanston, Illinois 60208-4070, U.S.A.severini{at}nwu.edu

Consider inference about a scalar parameter of interest {Theta} in the presence of a vector nuisance parameter. Inference about {Theta} is often based on a pseudolikelihood function. In this paper, the general problem of constructing a pseudo-loglikelihood function H({Theta}) is considered. Conditions are given under which H has the same properties as a genuine loglikelihood function for a model without a nuisance parameter. When these conditions are satisfied to a given order of approximation, H is said to be a jth-order local loglikelihood function. The theory of local loglikelihood functions is developed and it is shown that second-order versions of these have a number of desirable properties. Several commonly used pseudolikelihood functions are studied from this point of view. One commonly used pseudolikelihood function is profile likelihood in which parameters other than {Theta} are replaced by their maximum likelihood estimates. A second aspect of the paper is to consider the use of other estimates in this context. Examples are given which suggest that inference about {Theta} may be improved if a method other than maximum likelihood is used, particularly when the number of other parameters is large relative to the sample size.

Key Words: Adjusted profile likelihood • Asymptotic theory • Local inference • Profile likelihood


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T. A. Severini
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[Abstract] [PDF]


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