Bayesian Inference for Extremes: Accounting for the Three Extremal Types |
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Authors: | Email author" target="_blank">Alec?StephensonEmail author Jonathan?Tawn |
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Institution: | (1) Department of Statistics, Division of Economic and Financial Studies, Macquarie University, NSW 2109, Australia;(2) Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF, UK |
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Abstract: | The Extremal Types Theorem identifies three distinct types of extremal behaviour. Two different strategies for statistical
inference for extreme values have been developed to exploit this asymptotic representation. One strategy uses a model for
which the three types are combined into a unified parametric family with the shape parameter of the family determining the
type: positive (Fréchet), zero (Gumbel), and negative (negative Weibull). This form of approach never selects the Gumbel type
as that type is reduced to a single point in a continuous parameter space. The other strategy first selects the extremal type,
based on hypothesis tests, and then estimates the best fitting model within the selected type. Such approaches ignore the
uncertainty of the choice of extremal type on the subsequent inference. We overcome these deficiencies by applying the Bayesian
inferential framework to an extended model which explicitly allocates a non-zero probability to the Gumbel type. Application
of our procedure suggests that the effect of incorporating the knowledge of the Extremal Types Theorem into the inference
for extreme values is to reduce uncertainty, with the degree of reduction depending on the shape parameter of the true extremal
distribution and the prior weight given to the Gumbel type. |
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Keywords: | extreme value theory extremal types theorem generalized extreme value distribution Poisson process reversible jump Markov chain Monte Carlo |
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