A second-order adjustment to the profile likelihood in the case of a multidimensional parameter of interest

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Abstract

Inference in the presence of nuisance parameters is often carried out by using the χ2-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.

Original languageEnglish
Pages (from-to)653-665
Number of pages13
JournalJournal of the Royal Statistical Society. Series B: Statistical Methodology
Volume59
Issue number3
DOIs
Publication statusPublished - 1997
Externally publishedYes

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Profile Likelihood
Adjustment
Nuisance Parameter
Small Sample
Score Statistic
Likelihood Ratio Statistic
Likelihood Function
Vanish
Likelihood
Approximation

Cite this

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abstract = "Inference in the presence of nuisance parameters is often carried out by using the χ2-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.",
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AU - Stern, Steven E.

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N2 - Inference in the presence of nuisance parameters is often carried out by using the χ2-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.

AB - Inference in the presence of nuisance parameters is often carried out by using the χ2-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.

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