7,108 research outputs found
Extreme value copula estimation based on block maxima of a multivariate stationary time series
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 infinity. 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.Comment: 34 page
On the occurrence times of componentwise maxima and bias in likelihood inference for multivariate max-stable distributions
Full likelihood-based inference for high-dimensional multivariate extreme
value distributions, or max-stable processes, is feasible when incorporating
occurrence times of the maxima; without this information, -dimensional
likelihood inference is usually precluded due to the large number of terms in
the likelihood. However, some studies have noted bias when performing
high-dimensional inference that incorporates such event information,
particularly when dependence is weak. We elucidate this phenomenon, showing
that for unbiased inference in moderate dimensions, dimension should be of
a magnitude smaller than the square root of the number of vectors over which
one takes the componentwise maximum. A bias reduction technique is suggested
and illustrated on the extreme value logistic model.Comment: 7 page
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