Journal of Statistical Theory and Practice

, Volume 6, Issue 3, pp 566–579 | Cite as

Bayesian Quantiles of Extremes

  • Branko MiladinovicEmail author
  • Chris P. Tsokos


Extreme value distributions are increasingly being applied in biomedical literature to model unusual behavior or rare events. Two popular methods that are used to estimate the location and scale parameters of the type I extreme value (or Gumbel) distribution, namely, the empirical distribution function and the method of moments, are not optimal, especially for small samples. Additionally, even with the more robust maximum likelihood method, it is difficult to make inferences regarding outcomes based on estimates of location and scale parameters alone. Quantile modeling has been advocated in statistical literature as an intuitive and comprehensive approach to inferential statistics. We derive Bayesian estimates of the Gumbel quantile function by utilizing the Jeffreys noninformative prior and Lindley approximation procedure. The advantage of this approach is that it utilizes information on the prior distribution of parameters, while making minimal impact on the estimated posterior distribution. The Bayesian and maximum likelihood estimates are compared using numerical simulation. Numerical results indicate that Bayesian quantile estimates are closer to the true quantiles than their maximum likelihood counterparts. We illustrate the method by applying the estimates to published extreme data from the analysis of streak artifacts on computed tomography (CT) images.



Bayesian inference Extreme value distribution Lindley procedure noninformative prior Quantiles 


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  1. Abrams, A. M., K. Kleinman, and M. Kulldorff. 2010. Gumbel based p-value approximations for spatial scan statistics. Int. J. Health Geogr. 9; 61.CrossRefGoogle Scholar
  2. Ahsanullah, M. 1990. Estimation of the parameters of the Gumbel distribution based on the m record values. Comput. Stat. Qu., 6, 231–239.MathSciNetzbMATHGoogle Scholar
  3. Ahsanullah, M. 1991. Inference and prediction of the Gumbel distribution based on record values. Pakistan. J. Stat., 7(3)B, 53–62.MathSciNetzbMATHGoogle Scholar
  4. Ali Mousa, M. A., Z. F. Jaheen, and A. A. Ahmad. 2001. Bayesian estimation, prediction, and characterization for the Gumbel model based on records. Statistics, 36(1), 65–74.MathSciNetCrossRefGoogle Scholar
  5. Bastien, O., and E. Marechel. 2008. Evolution of biological sequences implies and extreme value distribution of type I for both global and local pairwise alignment scores. BMC Bioinformatics, 9, 332.CrossRefGoogle Scholar
  6. Fisher, R. A., and L. H. C. Tippett. 1928. Limiting forms of the frequency distribution of the largest or smallest member of a sample. Proc. Cambridge Philosophical Society W24, 180–290.Google Scholar
  7. Gumbel, E. J. 1958. Statistics of extremes. New York, Columbia University Press.zbMATHGoogle Scholar
  8. Haan, L. D., and A. Ferreira. 2006. Extreme value theory: An introduction. Springer Series in Operations Research. New York, Springer.CrossRefGoogle Scholar
  9. Hugueny, S., D. A. Clifton, and L., Tarassenko. 2010. Probabilistic patient monitoring with multivariate, multimodal extreme value theory. Commun. Comput. and Information Sci. Invited article, from IEEE Biomedical Engineering Systems and Technologies. 127, 199–211.CrossRefGoogle Scholar
  10. Imai, K., et al. 2002. A detection method for streak artifacts and radiological noise in a non-uniform region in a CT image. Phys. Med., 26(3), 157–165.CrossRefGoogle Scholar
  11. Imai, K., et al. 2007. Analysis of streak artifacts on CT images using statistics of extremes. Br. J. Radiol., 80(959), 911–918.CrossRefGoogle Scholar
  12. Imai, K., et al. 2009. Statistical characteristics of streak artifacts on CT images: Relationship between streak artifacts and mA s values. Med. Phys., 36(2), 492–499.CrossRefGoogle Scholar
  13. Ivanek, R., et al., 2008. Extreme value theory in analysis of differential expression in microarrays where either only up- or down-regulated genes are relevant or expected. Genet. Res. (Cambr.) 90(4), 347–361.CrossRefGoogle Scholar
  14. Kimball B. F., 1960. On the choice plotting positions on probability paper. JASA, 55(291), 546–560.CrossRefGoogle Scholar
  15. Leong, Y. P., J. W. Sleigh, and J. M. Torrance. 2002. Extreme value theory applied to postoperative breathing patterns. Br. J. Anaesth., 88(1), 61–64.CrossRefGoogle Scholar
  16. Parzen, E., 1979. Nonparametric statistical data modeling. JASA, 74, 105–131.MathSciNetCrossRefGoogle Scholar
  17. Parzen, E. 2004. Quantile probability and statistical data modeling. Stat. Sci., 19, 652–662.MathSciNetCrossRefGoogle Scholar
  18. Parzen, E. 2009. Quantiles, conditional quantiles, confidence quantiles for p, logodds(p). Commun. Stat. Theory Methods, 38, 3048–3058.MathSciNetCrossRefGoogle Scholar
  19. Sariyar, M., A. Borg, and K. Pommerening. 2011. Controlling false match rates in record linkage using extreme value theory. J. Biomed. Inform. 44(4), 648–654.CrossRefGoogle Scholar
  20. Shorak, G. R. and J. A. Wellner. 1986. Empirical processes with applications to statistics. NewYork, Wiley.Google Scholar

Copyright information

© Grace Scientific Publishing 2012

Authors and Affiliations

  1. 1.Center for Evidence Based Medicine and Health Outcomes ResearchUniversity of South FloridaTampaUSA
  2. 2.Department of Mathematics and StatisticsUniversity of South FloridaTampaUSA

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