Abstract
This paper presents an adaptive version of the Hill estimator based on Lespki's model selection method. This simple data-driven index selection method is shown to satisfy an oracle inequality and is checked to achieve the lower bound recently derived by Carpentier and Kim. In order to establish the oracle inequality, we derive non-asymptotic variance bounds and concentration inequalities for Hill estimators. These concentration inequalities are derived from Talagrand's concentration inequality for smooth functions of independent exponentially distributed random variables combined with three tools of Extreme Value Theory: the quantile transform, Karamata's representation of slowly varying functions, and Renyi 's charac-terisation of the order statistics of exponential samples. The performance of this computationally and conceptually simple method is illustrated using Monte-Carlo simulations.
Highlights
The basic questions faced by Extreme Value Analysis consist in estimating the probability of exceeding a threshold that is larger than the sample maximum and estimating a quantile of an order that is larger than 1 minus the reciprocal of the sample size, that is making inferences on regions that lie outside the support of the empirical distribution
In order to face these challenges in a sensible framework, Extreme Value Theory (EVT) assumes that the sampling distribution F satisfies a regularity condition
The sampling distribution F is said to belong to the maxdomain of attraction of a Frechet distribution with index γ > 0 (abbreviated in F ∈ MDA(γ)) and γ is called the extreme value index
Summary
The basic questions faced by Extreme Value Analysis consist in estimating the probability of exceeding a threshold that is larger than the sample maximum and estimating a quantile of an order that is larger than 1 minus the reciprocal of the sample size, that is making inferences on regions that lie outside the support of the empirical distribution. We combine Talagrand’s concentration inequality for smooth functions of independent exponentially distributed random variables (Theorem 2.15) with three traditional tools of EVT: the quantile transform, Karamata’s representation for slowly varying functions, and Renyi’s characterisation of the joint distribution of order statistics of exponential sample√s.
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