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METRON

, Volume 76, Issue 3, pp 305–320 | Cite as

Likelihood, credible and confidence intervals for parameters in complex models

  • Murray Aitkin
Article
  • 40 Downloads

Abstract

Recent papers have discussed general procedures with complex models to obtain confidence interval statements for a parameter of interest in the presence of nuisance parameters. This paper discusses the role of the likelihood in these procedures, and points out the simplicity of the Bayesian credible interval approach to the same models.

Keywords

Likelihood Pivot Confidence distribution Asymptotic coverage Non-informative prior 

References

  1. 1.
    Aitkin, M.: Profile likelihood. In: Encyclopedia of Biostatistics, pp. 3534–3536. Wiley, New York (1998)Google Scholar
  2. 2.
    Aitkin, M.: Bayesian bootstrap analysis of regression in finite population survey data with stratification and clustering. J. Off. Stat. 24, 21–51 (2008)Google Scholar
  3. 3.
    Aitkin, M.: Statistical inference: an integrated Bayesian/likelihood approach. CRC Press, Boca Raton (2010)CrossRefGoogle Scholar
  4. 4.
    Angus, J.E.: Bootstrapping a universal pivot when nuisance parameters are estimated. Am. Stat. 70(1), 100–107 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Berger, J.O., Bernardo, J.M.: On the development of reference priors (with discussion). In Bayesian statistics, vol. 4, pp. 35–60. Oxford University Press, London (1992)Google Scholar
  6. 6.
    Cox, D.R.: Principles of statistical inference. Cambridge University Press, Cambridge (2016)zbMATHGoogle Scholar
  7. 7.
    Datta, G.S., Sweeting, T.J.: Probability matching priors. In: Dey, D.K., Rao, C.R. (eds.) Handbook of statistics, 25: Bayesian thinking: modeling computation, pp. 91–114. Elsevier, Amsterdam (2005)CrossRefGoogle Scholar
  8. 8.
    Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., Rubin, D.B.: Bayesian Data Analysis, 3rd edn. CRC Press, Boca Raton (2014)zbMATHGoogle Scholar
  9. 9.
    Lunn, D., Jackson, C., Best, N., Thomas, A., Spiegelhalter, D.: The BUGS Book. CRC Press, Boca Raton (2013)zbMATHGoogle Scholar
  10. 10.
    Ohlsson, U.: Confidence intervals for the mean of a lognormal distribution. Journal of Statistics Education 13. (2005) http://ww2.amstat.org/publications/jse/v13n1/olsson.html. Accessed 1 Sept 2018
  11. 11.
    Rolke, W.A., López, A.M., Conrad, J.: Limits and confidence intervals in the presence of nuisance parameters. Nucl. Instrum. Meth. A 551, 493–503 (2005)CrossRefGoogle Scholar
  12. 12.
    Schweder, T., Hjort, N.L.: Confidence, likelihood, probability. Cambridge University Press, Cambridge (2016)CrossRefGoogle Scholar
  13. 13.
    Sen, B., Walker, M., Woodroofe, M.: On the unified method with nuisance parameters. Stat. Sinica 19, 301–314 (2009)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Tanner, M.A.: Tools for statistical inference: methods for the exploration of posterior distributions and likelihood functions, 3rd edn. Springer, New York (1993)CrossRefGoogle Scholar
  15. 15.
    Tanner, M.A., Wong, W.H.: The calculation of posterior distributions by data augmentation (with discussion). J Am Stat. Assoc. 82, 528–550 (1987)CrossRefGoogle Scholar
  16. 16.
    Welch, B.L.: On confidence limits and sufficiency with particular reference to parameters of location. Ann. Math. Stat. 10, 58–69 (1939)CrossRefGoogle Scholar

Copyright information

© Sapienza Università di Roma 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of MelbourneMelbourneAustralia

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