Skip to main content
SpringerLink
Log in

Simulation Methods

  • Chapter
  • First Online:
All of Statistics

Part of the book series: Springer Texts in Statistics ((STS))

  • 580k Accesses

Abstract

In this chapter we will show how simulation can be used to approximate integrals. Our leading example is the problem of computing integrals in Bayesian inference but the techniques are widely applicable. We will look at three integration methods: (i) basic Monte Carlo integration, (ii) importance sampling, and (iii) Markov chain Monte Carlo (MCMC).

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 64.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 79.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  • Agresti, A. (1990).Categorical Data Analysis.Wiley.

    Google Scholar 

  • Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle.Second International Symposium on Information Theory 267–281.

    Google Scholar 

  • Anderson, T. W. (1984). AnIntroduction to Multivariate Statistical Analysis (Second Edition). Wiley.

    Google Scholar 

  • Barron, A., Schervish, M. J. and Wasserman, L. (1999). The consistency of posterior distributions in nonparametric problems.The Annals of Statistics27 536–561.

    Article  MathSciNet  Google Scholar 

  • Beecher, H. (1959).Measurement of Subjective Responses.Oxford University Press.

    Google Scholar 

  • Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing.Journal of the Royal Statistical Society,Series B,Methodological57 289–300.

    MATH  Google Scholar 

  • Reran, R. (2000). React scatterplot smoothers: Superefficiency through basis economy.Journal of the American Statistical Association95 155–171.

    Article  MathSciNet  Google Scholar 

  • Beran, R. and Dümbgen, L. (1998). Modulation of estimators and confidence sets.The Annals of Statistics26 1826–1856.

    Article  MathSciNet  Google Scholar 

  • Berger, J. and Wolpert, R. (1984). The Likelihood Principle. Institute of Mathematical Statistics.

    Google Scholar 

  • Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis(Second Edition). Springer-Verlag.

    Google Scholar 

  • Berger, J. O. and Delampady, M. (1987). Testing precise hypotheses (c/r: P335–352).Statistical Science2 317–335.

    Article  MathSciNet  Google Scholar 

  • Berliner, L. M. (1983). Improving on inadmissible estimators in the control problem.The Annals of Statistics11 814–826.

    Article  MathSciNet  Google Scholar 

  • Bickel, P. J. and Doxsum, K. A. (2000). Mathematical Statistics: Basic Ideas and Selected Topics, Vol. I (Second Edition).Prentice Hall.

    Google Scholar 

  • Billingsley, P. (1979). Probability and Measure. Wiley.

    Google Scholar 

  • Bishop, Y. M. M.. Fienberg, S. E. and Holland, P. W. (1975).Discrete Multivariate Analyses: Theory and Practice. MIT Press.

    Google Scholar 

  • Breiman, L. (1992).Probability. Society for Industrial and Applied Mathematics.

    Google Scholar 

  • Brinegar, C. S. (1963). Mark Twain and the Quintus Curtius Snodgrass letters: A statistical test of authorship.Journal of the American Statistical Association58 85–96.

    Article  Google Scholar 

  • Carlin, B. P. and Lows, T. A. (1996). Bayes and Empirical Bayes Methods for Data Analysis. Chapman & Hall.

    Google Scholar 

  • Casella, G. and Berger, R. L. (2002). Statistical Inference. Duxbury Press.

    Google Scholar 

  • Chaudhuri, P. and Marron, J. S. (1999). Sizer for exploration of structures in curves.Journal of the American Statistical Association94 807–823.

    Article  MathSciNet  Google Scholar 

  • Cox, D. and Lewis, P. (1966). The Statistical Analysis of Series of Events. Chapman & Hall.

    Google Scholar 

  • Cox, D. D. (1993). An analysis of Bayesian inference for nonparametric regression.The Annals of Statistics21 903–923.

    Article  MathSciNet  Google Scholar 

  • Davison, A. C. and Hinkley, D. V. (1997). Bootstrap Methods and Their Application. Cambridge University Press.

    Google Scholar 

  • Degroot, M. and Schervish, M. (2002). Probability and Statistics (Third Edition). Addison-Wesley.

    Google Scholar 

  • Devroye, L., Györfi, L. and Lugosi, G. (1996). AProbabilistic Theory of Pattern Recognition. Springer-Verlag.

    Google Scholar 

  • Diaconis, P. and Freedman, D. (1986). on inconsistent Bayes estimates of location.The Annals of Statistics14 68–87.

    Article  MathSciNet  Google Scholar 

  • Dobson, A. J. (2001). Anintroduction to generalized linear models. Chapman & Hall.

    Google Scholar 

  • Donoho, D. L. and Johnstone, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage.Biometrika81 425–455.

    Article  MathSciNet  Google Scholar 

  • Donoho, D. L. and Johnstone, I. M. (1995). Adapting to unknown smoothness via wavelet shrinkage.Journal of the American Statistical Association90 1200–1224.

    Article  MathSciNet  Google Scholar 

  • Donoho, D. L. and Johnstone, I. M. (1998). Minimax estimation via wavelet shrinkage.The Annals of Statistics26 879–921.

    Article  MathSciNet  Google Scholar 

  • Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1995). Wavelet shrinkage: Asymptopia? (Disc: p 337–369).Journal of the Royal Statistical Society,Series B,Methodological57 301–337.

    MATH  Google Scholar 

  • Dunsmore, I., Daly, F. ET AL. (1987).M345 Statistical Methods , Unit 9: Categorical Data. The Open University.

    Google Scholar 

  • Edwards, D. (1995).Introduction to graphical modelling. Springer-Verlag.

    Google Scholar 

  • Efromovich, S. (1999).Nonparametric Curve Estimation: Methods , Theory and Applications. Springer-Verlag.

    Google Scholar 

  • Efron, B. (1979). Bootstrap methods: Another look at the jackknife.The Annals of Statistics7 1–26.

    Article  MathSciNet  Google Scholar 

  • Efron, B., Tibshirani, R., Storey, J. D. and Tusher, V. (2001). Empirical Bayes analysis of a microarray experiment.Journal of the American Statistical Association96 1151–1160.

    Article  MathSciNet  Google Scholar 

  • Efron, B. and Tibshirani, R. J. (1993).An Introduction to the Bootstrap. Chapman & Hall.

    Google Scholar 

  • Ferguson, T. (1967).Mathematical Statistics: a Decision Theoretic Approach. Academic Press.

    Google Scholar 

  • Fisher, R. (1921). On the probable error of a coefficient of correlation deduced from a small sample.Metron1 1–32.

    Google Scholar 

  • Freedman, D. (1999). Wald lecture: On the Bernstein-von Mises theorem with infinite-dimensional parameters.The Annals of Statistics27 1119–1141.

    MathSciNet  MATH  Google Scholar 

  • Friedman, J. H. (1997). On bias, variance, 0/1-loss, and the curse-ofdimensionality.Data Mining and Knowledge Discovery1 55–77.

    Article  Google Scholar 

  • Gelman, A., Carlin, J. B., Stern, H. S. and Rubin, D. B. (1995).Bayesian Data Analysis. Chapman & Hall.

    Google Scholar 

  • Ghosal, S., Ghosh, J. K. and Van Der Vaart, A. W. (2000). Convergence rates of posterior distributions.The Annals of Statistics28 500–531.

    Article  MathSciNet  Google Scholar 

  • Gilks, W. R., Richardson, S. and Spiegelhalter, D. J. (1998).Markov Chain Monte Carlo in Practice. Chapman & Hall.

    Google Scholar 

  • Grimmett, G. and Stirzaker, D. (1982).Probability and Random Processes. Oxford University Press.

    Google Scholar 

  • Guttorp, P. (1995).Stochastic Modeling of Scientific Data. Chapman & Hall.

    Google Scholar 

  • Hardle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998).Wavelets, Approximation, and Statistical Applications. Springer-Verlag.

    Google Scholar 

  • Hall, P. (1992).The Bootstrap and Edgeworth Expansion. Springer-Verlag.

    Google Scholar 

  • Halverson, N., Leitch, E., Pryke, C., Kovac, J., Carlstrom, J., Holzapfel, W., Dragovan, M., Cartwright, J., Mason, B., Padin, S., Pearson, T., Shepherd, M. and Readhead, A. (2002). DASI first results: A measurement of the cosmic microwave background angular power spectrum.Astrophysics Journal568 38–45.

    Article  Google Scholar 

  • Hardle, W. (1990).Applied nonparametric regression. Cambridge University Press.

    Google Scholar 

  • Hastie, T., Tibshirani, R. and Friedman, J. H. (2001).The Elements of Statistical Learning: Data Mining , Inference , and Prediction. Springer-Verlag.

    Google Scholar 

  • Herbich, R. (2002). Learning Kernel Classifiers: Theory and Algorithms. MIT Press.

    Google Scholar 

  • Johnson, R. A. and Wichern, D. W. (1982).Applied Multivariate Statistical Analysis. Prentice-Hall.

    Google Scholar 

  • Johnson, S. and Johnson, R. (1972).New England Journal of Medicine287 1122–1125.

    Article  Google Scholar 

  • Jordan, M. (2004).Graphical models. In Preparation.

    Google Scholar 

  • Karr, A. (1993).Probability. Springer-Verlag.

    Google Scholar 

  • Kass, R. E. and Raftery, A. E. (1995). Bayes factors.Journal of the American Statistical Association90 773–795.

    Article  MathSciNet  Google Scholar 

  • Kass, R. E. and Wasserman, L. (1996). The selection of prior distributions by formal rules (corr: 1998 v93 p 412).Journal of the American Statistical Association91 1343–1370.

    Article  Google Scholar 

  • Larsen, R. J. and Marx, M. L. (1986).An Introduction to Mathematical Statistics and Its Applications (Second Edition). Prentice Hall.

    Google Scholar 

  • Lauritzen, S. L. (1996).Graphical Models. Oxford University Press.

    Google Scholar 

  • Lee, A. T. et al. (2001). A high spatial resolution analysis of the maxima-1 cosmic microwave background anisotropy data.Astrophys. J.561 L1 - L6.

    Article  Google Scholar 

  • Lee, P. M. (1997).Bayesian Statistics: An Introduction. Edward Arnold.

    Google Scholar 

  • Lehmann, E. L. (1986).Testing Statistical Hypotheses (Second Edition). Wiley.

    Google Scholar 

  • Lehmann, E. L. and Casella, G. (1998).Theory of Point Estimation. Springer-Verlag.

    Google Scholar 

  • Loader, C. (1999).Local regression and likelihood. Springer-Verlag.

    Google Scholar 

  • Marron, J. S. and Wand, M. P. (1992). Exact mean integrated squared error.The Annals of Statistics20 712–736.

    Article  MathSciNet  Google Scholar 

  • Morrison, A., Black, M., Lowe, C., Macmahon, B. and Yusa, S. (1973). Some international differences in histology and survival in breast cancer.International Journal of Cancer11 261–267.

    Article  Google Scholar 

  • Netterfield, C. B. et al. (2002). A measurement by boomerang of multiple peaks in the angular power spectrum of the cosmic microwave background.Astrophys. J.571 604–614.

    Article  Google Scholar 

  • Ogden, R. T. (1997).Essential Wavelets for Statistical Applications and Data Analysis. Birkhäuser.

    Google Scholar 

  • Pearl, J. (2000).Casuality: models , reasoning , and inference. Cambridge University Press.

    Google Scholar 

  • Phillips, D. and King, E. (1988). Death takes a holiday: Mortality surrounding major social occasions.Lancet2 728–732.

    Article  Google Scholar 

  • Phillips, D. and Smith, D. (1990). Postponement of death until symbolically meaningful occasions.Journal of the American Medical Association263 1947–1961.

    Article  Google Scholar 

  • Quenouille, M. (1949). Approximate tests of correlation in time series.Journal of the Royal Statistical Society B11 18–84.

    MathSciNet  MATH  Google Scholar 

  • Rice, J. A. (1995).Mathematical Statistics and Data Analysis (Second Edition). Duxbury Press.

    Google Scholar 

  • Robert, C. P. (1994).The Bayesian Choice: A Decision-theoretic Motivation. Springer-Verlag.

    Google Scholar 

  • Robert, C. P. and Casella, G. (1999).Monte Carlo Statistical Methods. Springer-Verlag.

    Google Scholar 

  • Robins, J., Scheines, R., Spirtes, P. and Wasserman, L. (2003). Uniform convergence in causal inference.Biometrika (to appear).

    Google Scholar 

  • Robins, J. M. and Ritov, Y. (1997). Toward a curse of dimensionality appropriate (CODA) asymptotic theory for semi-parametric models.Statistics in Medicine16 285–319.

    Article  Google Scholar 

  • Rosenbaum, P. (2002).Observational Studies. Springer-Verlag.

    Google Scholar 

  • Rousseauw, J., Du Plessis, J., Benade, A., Jordaan, P., Kotze, J., Jooste, P. and Ferreira, J. (1983). Coronary risk factor screening in three rural communities.South African Medical Journal64 430–436.

    Google Scholar 

  • Schervish, M. J. (1995).Theory of Statistics. Springer-Verlag.

    Google Scholar 

  • Scholkopf, B. and Smola, A. (2002).Learning with Kernels: Support Vector Machines , Regularization , Optimization , and Beyond. MIT Press.

    Google Scholar 

  • Schwarz, G. (1978). Estimating the dimension of a model.The Annals of Statistics6 461–464.

    Article  MathSciNet  Google Scholar 

  • Scott, D., Lotto, A., Cole, J. and Gorry, G. (1978). Plasma lipids as collateral risk factors in coronary artery disease: a study of 371 males with chest pain.Journal of Chronic Diseases31 337–345.

    Article  Google Scholar 

  • Scott, D. W. (1992).Multivariate Density Estimation: Theory , Practice , and Visualization. Wiley.

    Google Scholar 

  • Shao, J. and Tu, D. (1995).The Jackknife and Bootstrap (German). Springer-Verlag.

    Google Scholar 

  • Shen, X. and Wasserman, L. (2001). Rates of convergence of posterior distributions.The Annals of Statistics29 687–714.

    Article  MathSciNet  Google Scholar 

  • Shorack, G. R. and Wellner, J. A. (1986).Empirical Processes With Applications to Statistics. Wiley.

    Google Scholar 

  • Silverman, B. W. (1986).Density Estimation for Statistics and Data Analysis. Chapman & Hall.

    Google Scholar 

  • Spirtes, P., Glymour, C. N. and Scheines, R. (2000).Causation , prediction , and search. MIT Press.

    Google Scholar 

  • Taylor, H. M. and Karlin, S. (1994).An Introduction to Stochastic Modeling. Academic Press.

    Google Scholar 

  • Van Der Laan, M. and Robins, J. (2003).Unified Methods for Censored Longitudinal Data and Causality. Springer Verlag.

    Google Scholar 

  • Van Der Vaart, A. W. (1998).Asymptotic Statistics. Cambridge University Press.

    Google Scholar 

  • Vapnik, V. N. (1998).Statistical Learning Theory. Wiley.

    Google Scholar 

  • Weisberg, S. (1985).Applied Linear Regression. Wiley.

    Google Scholar 

  • Whittaker, J. (1990).Graphical Models in Applied Multivariate Statistics. Wiley.

    Google Scholar 

  • Wright, S. (1934). The method of path coefficients.The Annals of Mathematical Statistics5 161–215.

    Article  Google Scholar 

  • Zhao, L. H. (2000). Bayesian aspects of some nonparametric problems.The Annals of Statistics28 532–552.

    Article  MathSciNet  Google Scholar 

  • Zheng, X. and Loh, W.-Y. (1995). Consistent variable selection in linear models.Journal of the American Statistical Association90 151–156.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, Carnegie Mellon University, Baker Hall 228A, 15213-3890, Pittsburgh, PA, USA

    Larry Wasserman

Authors
  1. Larry Wasserman
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer Science+Business Media New York

About this chapter

Cite this chapter

Wasserman, L. (2004). Simulation Methods. In: All of Statistics. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21736-9_24

Download citation

  • DOI: https://doi.org/10.1007/978-0-387-21736-9_24

  • Published:

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4419-2322-6

  • Online ISBN: 978-0-387-21736-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation