Skip to main content
SpringerLink
Log in

Efficient simulated maximum likelihood with an application to online retailing

  • Published:
Statistics and Computing Aims and scope Submit manuscript

Abstract

Simulated maximum likelihood estimates an analytically intractable likelihood function with an empirical average based on data simulated from a suitable importance sampling distribution. In order to use simulated maximum likelihood in an efficient way, the choice of the importance sampling distribution as well as the mechanism to generate the simulated data are crucial. In this paper we develop a new heuristic for an automated, multistage implementation of simulated maximum likelihood which, by adaptively updating the importance sampler, approximates the (locally) optimal importance sampling distribution. The proposed approach also allows for a convenient incorporation of quasi-Monte Carlo methods. Quasi-Monte Carlo methods produce simulated data which can significantly increase the accuracy of the likelihood-estimate over regular Monte Carlo methods. Several examples provide evidence for the potential efficiency gain of this new method. We apply the method to a computationally challenging geostatistical model of online retailing.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Booth J. G. and Hobert J. P. 1999. Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm. Journal of the Royal Statistical Society B 61: 265–285.

    Article  MATH  Google Scholar 

  • Booth J. G., Hobert J. P., and Jank W. 2001. A survey of Monte Carlo algorithms for maximizing the likelihood of a two-stage hierarchical model. Statistical Modelling 1: 333–349.

    Article  MATH  Google Scholar 

  • Bouleau N. and Lépingle D. 1994. Numerical Methods for Stochastic Processes. Wiley, New York.

    MATH  Google Scholar 

  • Caffo B. S., Booth J. G., and Davison A. C. 2002. Empirical supremum rejection sampling. Biometrika 89: 745–754.

    Article  MathSciNet  MATH  Google Scholar 

  • Caflisch R., Morokoff W., and Owen A. 1997. Valuation of mortgage-backed securities using brownian bridges to reduce effective dimension. Journal of Computational Finance 1: 27–46.

    Google Scholar 

  • Dempster A. P., Laird N. M., and Rubin D. B. 1977. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society B 39: 1–22.

    MathSciNet  MATH  Google Scholar 

  • Diggle P. J., Tawn J. A., and Moyeed R. A. 1998. Model-based geostatistics. Journal of the Royal Statistical Society A 47: 299–350.

    Article  MathSciNet  MATH  Google Scholar 

  • Doornik J. A. 2001. Ox: Object Oriented Matrix Programming. Timberlake, London.

    Google Scholar 

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

    MATH  Google Scholar 

  • Floudas C. and Pardalos P., editors 2001. Encyclopedia of Optimization. Kluwer Academic Publishers.

  • Halton J. H. 1960. On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numerische Mathematik 2: 84–90.

    Article  MATH  MathSciNet  Google Scholar 

  • Hobert J. P. 2000. Hierarchical models: A current computational perspective. Journal of the American Statistical Association 95: 1312–1316.

    MATH  MathSciNet  Google Scholar 

  • Jank W. and Booth J. G. 2003. Efficiency of Monte Carlo EM and Simulated Maximum Likelihood in two-stage hierarchical models. Journal of Computational and Graphical Statistics 12: 214–229.

    Article  MathSciNet  Google Scholar 

  • Kuk A. Y. C. 1999. Laplace importance sampling for generalized linear mixed models. Journal of Statistical Computation and Simulation 63: 143–158.

    MATH  MathSciNet  Google Scholar 

  • L’Ecuyer P. 1990. A unified view of the IPA, SF, and LR gradient estimation techniques. Management Science 36(11): 1364–1383.

    Article  MATH  Google Scholar 

  • L’Ecuyer P. 1993. Two Approaches for Estimating the Gradient in Functional Form. In Proceedings of the 1993 Winter Simulation Conference, pages 338–346. IEEE Press.

  • L’Ecuyer P. 1994. Efficiency improvement via variance reduction. In Proceedings of the 1994 Winter Simulation Conference, pages 122–132. IEEE Press.

  • L’Ecuyer P. and Lemieux C. 2002. Recent advances in randomized Quasi-Monte Carlo Methods. In Dror M., L’Ecuyer P., and Szidarovszki F., editors, Modeling Uncertainty: An Examination of Stochastic Theory, Methods, and Applications, pages 419–474. Kluwer Academic Publishers.

  • L’Ecuyer P. 2004. SSJ: A Java Library for Stochastic Simulation. Software user’ guide. Available at http://www.iro.umontreal.ca/lecuyer.

  • Lemieux C. and L’Ecuyer P. 1998. Efficiency improvement by lattice rules for pricing Asian options. In Proceedings of the 1998 Winter Simulation Conference, pages 579–586. IEEE Press.

  • Liao J. G. 1998. Variance reduction in Gibbs samplers using quasi-random numbers. Journal of Computational and Graphical Statistics 7: 253–266.

    Google Scholar 

  • McCullagh P. and Nelder J. A. 1989. Generalized Linear Models. Chapman & Hall, New York.

    MATH  Google Scholar 

  • McCulloch C. E. 1997. Maximum likelihood algorithms for generalized linear mixed models. Journal of the American Statistical Association 92: 162–170.

    MATH  MathSciNet  Google Scholar 

  • McCulloch C. E. and Searle, S. R. 2001. Generalized, linear, and mixed models. John Wiley and Sons.

  • Meng X.-L. and Rubin D. B. 1994. On the Global and Componentwise Rates of Convergence of the EM algorithm. Linear Algebra and its Applications 199: 413–425.

    Article  MathSciNet  MATH  Google Scholar 

  • Niederreiter H. 1992. Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia.

  • Owen A. B. 1998. Monte Carlo extension of Quasi-Monte Carlo. In 1998 Winter Simulation Conference Proceedings, pages 571–577. Springer, New York.

  • Owen A. B. and Zhou Y. 2000. Safe and effective importance sampling. Journal of the American Statistical Association 95: 135–143.

    MathSciNet  MATH  Google Scholar 

  • Owen A. B. and Tribble S.D. 2005. A quasi-Monte Carlo Metropolis algorithm. Proceedings of the National Academy of Sciences (to appear).

  • Pagès G. 1992. Van der Corput sequences, Kakutani transforms and one-dimensional numerical integration. Journal of Computational and Applied Mathematics 44: 21–39.

    Article  MATH  MathSciNet  Google Scholar 

  • Searle S. R., Casella G., and McCulloch C. E. 1992. Variance Components. Wiley, New York.

    MATH  Google Scholar 

  • Wang X. and Hickernell F. J. 2000. Randomized Halton sequences. Mathematical and Computer Modelling 32: 887–899.

    Article  MathSciNet  MATH  Google Scholar 

  • Wang X. and Fang K.-T. 2003. The effective dimension and quasi-Monte Carlo integration. Journal of Complexity 19: 101–124.

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang H. 2002. On estimation and prediction for spatial generalized linear mixed models. Biometrics 58: 129–136.

    Article  MathSciNet  Google Scholar 

  • Zhang P. 1996. Nonparametric importance sampling. Journal of the American Statistical Association 91: 1245–1253.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Decision and Information Technologies, The Robert H. Smith School of Business, University of Maryland, 4322 Van Munching Hall, College Park, MD, 20742-1815

    Wolfgang Jank

Authors
  1. Wolfgang Jank
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wolfgang Jank.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jank, W. Efficient simulated maximum likelihood with an application to online retailing. Stat Comput 16, 111–124 (2006). https://doi.org/10.1007/s11222-006-6890-9

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11222-006-6890-9

Keywords

Search

Navigation