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A Second‐order Adjustment to the Profile Likelihood in the Case of a Multidimensional Parameter of Interest

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  • Steven E. Stern

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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  • Steven E. Stern, 1997. "A Second‐order Adjustment to the Profile Likelihood in the Case of a Multidimensional Parameter of Interest," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 59(3), pages 653-665.
    • Handle: RePEc:bla:jorssb:v:59:y:1997:i:3:p:653-665
    DOI: 10.1111/1467-9868.00089
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    Cited by:

    1. Cysneiros, Audrey H.M.A. & Cribari-Neto, Francisco & Araújo Jr., Carlos A.G., 2008. "On Birnbaum-Saunders inference," Computational Statistics & Data Analysis, Elsevier, vol. 52(11), pages 4939-4950, July.
    2. Céline Cunen & Nils Lid Hjort, 2022. "Combining information across diverse sources: The II‐CC‐FF paradigm," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(2), pages 625-656, June.
    3. Kakizawa, Yoshihide, 2017. "Third-order average local powers of Bartlett-type adjusted tests: Ordinary versus adjusted profile likelihood," Journal of Multivariate Analysis, Elsevier, vol. 153(C), pages 98-120.
    4. Luigi Pace & Alessandra Salvan & Laura Ventura, 2011. "Adjustments of profile likelihood through predictive densities," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(5), pages 923-937, October.
    5. Kim, Mi-Ok & Zhou, Mai, 2008. "Empirical likelihood for linear models in the presence of nuisance parameters," Statistics & Probability Letters, Elsevier, vol. 78(12), pages 1445-1451, September.
    6. Ib Thomsen & Li-Chun Zhang & Joseph Sexton, 2000. "Markov Chain Generated Profile Likelihood Inference under Generalized Proportional to Size Non-ignorable Non-response," Discussion Papers 274, Statistics Norway, Research Department.

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