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Statistics and Computing

, Volume 15, Issue 3, pp 189–196 | Cite as

Importance sampling with the generalized exponential power density

  • Alain DesgagnéEmail author
  • Jean-Françcois Angers
Article

Abstract

In this paper, the generalized exponential power (GEP) density is proposed as an importance function in Monte Carlo simulations in the context of estimation of posterior moments of a location parameter. This density is divided in five classes according to its tail behaviour which may be exponential, polynomial or logarithmic. The notion of p-credence is also defined to characterize and to order the tails of a large class of symmetric densities by comparing their tails to those of the GEP density.

The choice of the GEP density as an importance function allows us to obtain reliable and effective results when p-credences of the prior and the likelihood are defined, even if there are conflicting sources of information. Characterization of the posterior tails using p-credence can be done. Hence, it is possible to choose parameters of the GEP density in order to have an importance function with slightly heavier tails than the posterior. Simulation of observations from the GEP density is also addressed.

Keywords

importance sampling credence heavy tail density numerical integration Monte Carlo 

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References

  1. Angers J.-F. 1992. Use of Student-t prior for the estimation of normal means: A computational approach, in Bayesian Statistic IV. Bernardo J.M., Berger J.O., David A.P., and Smith A.F.M. (Eds.), Oxford University Press, New York, pp. 567–575.Google Scholar
  2. Angers J.-F. 1996. Protection against outliers using a symmetric stable law prior, in IMS Lecture Notes—Monograph Series Vol. 29, pp. 273–283.Google Scholar
  3. Angers J.-F. 2000. P-Credence and outliers. Metron 58: 81–108.Google Scholar
  4. Angers J.-F. and Berger J.O. 1991. Robust hierarchical Bayes estimation of exchangeable means. The Canadian Journal of Statistics 19: 39–56.Google Scholar
  5. Box G. and Tiao G. 1962. A further look at robustness via Bayes’s theorem. Biometrika 49: 419–432.Google Scholar
  6. Carlin B. and Polson N. 1991. Inference for nonconjugate Bayesian models using the Gibbs sampler. The Canadian Journal of Statistics 19: 399–405.Google Scholar
  7. Desgagné A. and Angers J.-F. 2003. Computational aspect of the generalized exponential power density. Tech. rep., Report No. CRM-2918, Université de Montréal (http://www. crm.umontreal.ca/pub/Rapports/2900-2999/2918.pdf).Google Scholar
  8. Fan T.H. and Berger J.O. 1992. Behaviour of the posterior distribution and inferences for a normal means with t prior distributions. Statistics & Decisions 10: 99–120.Google Scholar
  9. Geweke J. 1994. Priors for macroeconomic time series and their applications. Econometric Theory 10: 609–632.MathSciNetGoogle Scholar
  10. Johnson N.L., Kotz S., and Balakrishnan N. 1994. Continuous Univariate Distributions, (2nd edn.), Wiley, New York, Vol. 1.Google Scholar
  11. Meinhold R. and Singpurwalla N. 1989. Robustification of Kalman filter models. Journal of the American Statistical Association 84: 479–486.MathSciNetGoogle Scholar
  12. O’hagan A. 1990. Outliers and credence for location parameter inference. Journal of the American Statistical Association 85: 172–176.Google Scholar
  13. Reiss R.-D. and Thomas M. 1997. Statistical Analysis of Extreme Values. Birkhauser Verlag, Basel.Google Scholar
  14. Robert C. 1996. Méthodes de Monte Carlo par Chaînes de Markov. Economica, Paris.Google Scholar
  15. Ross S.M. 1997. Simulation, (2nd edn.), Academic Press, San Diego.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Département de mathématiques et de statistiqueUniversité de MontréalMontréal

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