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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((1-\alpha )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-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.</p>

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Bayesian Inference for the Extreme Value Index

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  • 出版模式：

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  • 发表日期：

    2026年7月2日

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  • Glenn Almelor,
  • Subhashis Ghosal

摘要

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 \((1-\alpha )\) ( 1 - α ) -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.