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A Second‐order Adjustment to the Profile Likelihood in the Case of a Multidimensional Parameter of Interest
Author(s) -
Stern Steven E.
Publication year - 1997
Publication title -
journal of the royal statistical society: series b (statistical methodology)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 6.523
H-Index - 137
eISSN - 1467-9868
pISSN - 1369-7412
DOI - 10.1111/1467-9868.00089
Subject(s) - likelihood function , nuisance parameter , inference , likelihood ratio test , statistic , empirical likelihood , restricted maximum likelihood , parametric statistics , statistics , likelihood principle , parametric model , mathematics , econometrics , function (biology) , statistical inference , estimating equations , computer science , maximum likelihood , artificial intelligence , quasi maximum likelihood , evolutionary biology , biology , estimator
Inference in the presence of nuisance parameters is often carried out by using the χ 2 ‐approximation to the profile likelihood ratio statistic. However, in small samples, the accuracy of such procedures may be poor, in part because the profile likelihood does not behave as a true likelihood, in particular having a profile score bias and information bias which do not vanish. To account better for nuisance parameters, various researchers have suggested that inference be based on an additively adjusted version of the profile likelihood function. Each of these adjustments to the profile likelihood generally has the effect of reducing the bias of the associated profile score statistic. However, these adjustments are not applicable outside the specific parametric framework for which they were developed. In particular, it is often difficult or even impossible to apply them where the parameter about which inference is desired is multidimensional. In this paper, we propose a new adjustment function which leads to an adjusted profile likelihood having reduced score and information biases and is readily applicable to a general parametric framework, including the case of vector‐valued parameters of interest. Examples are given to examine the performance of the new adjusted profile likelihood in small samples, and also to compare its performance with other adjusted profile likelihoods.