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Biometrika Advance Access originally published online on December 3, 2007
Biometrika 2007 94(4):985-991; doi:10.1093/biomet/asm076
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© 2007 Biometrika Trust

Miscellanea

Importance Sampling Via the Estimated Sampler

Masayuki Henmi, Ryo Yoshida and Shinto Eguchi

Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106-8569, Japan henmi{at}ism.ac.jp yoshidar{at}ism.ac.jp eguchi{at}ism.ac.jp

Received for publication 1 January 2006. Revision received 1 May 2007.

Monte Carlo importance sampling for evaluating numerical integration is discussed. We consider a parametric family of sampling distributions and propose the use of the sampling distribution estimated by maximum likelihood. The proposed method of importance sampling using the estimated sampling distribution is shown to improve the asymptotic variance of the ordinary method using the true sampling distribution. The argument is closely related to the discussion of the paradox in Henmi & Eguchi (2004). We focus on a condition under which the estimated integration value obtained by the proposed method has asymptotic zero variance.

Key Words: Asymptotic variance zero • Monte Carlo integration • Nuisance parameter effect



References

    Eguchi S. A geometric look at nuisance parameter effect of local powers in testing hypothesis. Ann. Inst. Statist. Math. (1991) 43:245–60.[CrossRef]

    Evans M., Swartz T. Approximating Integrals via Monte Carlo and Deterministic Methods (2000) Oxford: Oxford University Press.

    Henmi M., Eguchi S. A paradox concerning nuisance parameters and projected estimating functions. Biometrika (2004) 91:929–41.[Abstract/Free Full Text]

    Marriott P. On the local geometry of mixture models. Biometrika (2002) 89:77–93.[Abstract/Free Full Text]

    Murata N. An integral representation of functions using three-layered networks and their approximation bounds. Neural Networks (1996) 9:947–56.[CrossRef][ISI][Medline]

    Oh M. S., Berger J. O. Integration of multimodal functions by Monte Carlo importance sampling. J. Am. Statist. Assoc. (1993) 88:450–6.[CrossRef][ISI]


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This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
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What's this?