Abstract
Suppose a model has parameter \(\theta =(\psi , \lambda )\), where \(\psi \) is the parameter of interest and \(\lambda \) is a nuisance parameter. The integrated likelihood method eliminates \(\lambda \) from the likelihood function \(L(\psi , \lambda )\) by integrating with respect to a weight function \(\pi (\lambda | \psi )\). The resulting integrated likelihood function \(\bar{L}(\psi )\) can be used for inference for \(\psi \). However, the analytical form for the integrated likelihood is not always available. This paper discusses 12 different approaches to computing the integrated likelihood. Some methods were originally developed for other computation purposes and they are modified to fit in the integrated likelihood framework. Methods considered include direct numerical integration methods such as Monte Carlo integration method, importance sampling, Laplace method; marginal likelihood computation methods; and methods for computing the marginal posterior density. Simulation studies and real data example are presented to evaluate and compare these methods empirically.
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Acknowledgments
The work of T. A. Severini was supported by NSF Grant DMS-1308009. This research was supported in part through the computational resources and staff contributions provided for the Social Sciences Computing cluster (SSCC) at Northwestern University. Recurring funding for the SSCC is provided by Office of the President, Weinberg College of Arts and Sciences, Kellogg School of Management, the School of Professional Studies, and Northwestern University Information Technology.
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Zhao, Z., Severini, T.A. Integrated likelihood computation methods. Comput Stat 32, 281–313 (2017). https://doi.org/10.1007/s00180-016-0677-z
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DOI: https://doi.org/10.1007/s00180-016-0677-z