Extreme value copula estimation based on block maxima of a multivariate stationary time series
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Date
2013-11-29
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Abstract
The core of the classical block maxima method consists of fitting an
extreme value distribution to a sample of maxima over blocks extracted
from an underlying series. In asymptotic theory, it is usually postulated
that the block maxima are an independent random sample of an extreme
value distribution. In practice however, block sizes are finite, so that the
extreme value postulate will only hold approximately. A more accurate
asymptotic framework is that of a triangular array of block maxima, the
block size depending on the size of the underlying sample in such a way
that both the block size and the number of blocks within that sample tend
to infi nity. The copula of the vector of componentwise maxima in a block
is assumed to converge to a limit, which, under mild conditions, is then
necessarily an extreme value copula. Under this setting and for absolutely
regular stationary sequences, the empirical copula of the sample of vectors
of block maxima is shown to be a consistent and asymptotically normal
estimator for the limiting extreme value copula. Moreover, the empirical
copula serves as a basis for rank-based, nonparametric estimation of the
Pickands dependence function of the extreme value copula. The results are
illustrated by theoretical examples and a Monte Carlo simulation study.
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Keywords
absolutely regular process, block maxima method, empirical copula process, extreme value copula, Pickands dependence function, stationary time series, weak convergence