Infill asymptotics for estimators of the integral of the extreme value index function of the Brown-Resnick processes
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We consider a max-stable process with a positive extreme value index function and an associated spectral process that is assumed to be a continuous exponential martingale. Both processes are regularly sampled over [0,1] at high frequency n, with n going to infinity. We provide an estimator of the integral of the extreme value index function over [0,t], 0<t<1, and study its infill asymptotics properties. In particular we prove that the estimator is consistent, but we obtain a biased central limit theorem without hope of a suitable bias correction. We then consider m iid sample-continuous stochastic processes that belong to the domain of attraction of the max-stable process and assume that these processes are also regularly sampled at rate 1/n. We show that the average values of the previous estimator built over the highest paths of these random elements can lead to an inconsistent estimator of the integral of the extreme value index function.
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