Bayesian Inference for the Extreme Value Index

Abstract A classical problem in the field of extreme value theory is the estimation of the extreme value index (EVI), which enables modeling of the maximum (or minimum) of a random sample from an unknown distribution. The EVI classifies the underlying probability distributions into three categories for the maximum domain of attraction: positive, negative, or zero. The Hill and Pickands estimators for the EVI are well known in the frequentist literature. In this paper, we propose a Bayesian method for inference on the EVI. We define a sequence of functionals of the distribution that approximates the EVI. The constructed functionals are different depending on whether the EVI is positive, negative, or zero. We assign a Dirichlet process prior on the distribution of the observations and use the induced posterior distribution of the sequence of functionals to make inferences on the EVI. We establish a Bernstein-von Mises Theorem for the posterior distribution of the EVI, which gives rise to the posterior contraction rate and asymptotic frequentist coverage of a Bayesian $$(1-\alpha )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>α</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -credible interval. Further, to test the hypothesis that the EVI is zero, we propose a test based on a modified Bayes factor. We use a power of the empirical likelihood and compute the marginal likelihood under the null hypothesis by the Hamiltonian Monte Carlo method. We show the effectiveness of the proposed Bayesian methods through a comprehensive simulation study.