Bayesian Quantiles of Extremes

Abstract

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.

Keywords:

• 62F15

Keywords:

• Bayesian inference
• Extreme value distribution
• Lindley procedure
• noninformative prior
• Quantiles