299 research outputs found
Asymptotics of empirical copula processes under non-restrictive smoothness assumptions
Weak convergence of the empirical copula process is shown to hold under the
assumption that the first-order partial derivatives of the copula exist and are
continuous on certain subsets of the unit hypercube. The assumption is
non-restrictive in the sense that it is needed anyway to ensure that the
candidate limiting process exists and has continuous trajectories. In addition,
resampling methods based on the multiplier central limit theorem, which require
consistent estimation of the first-order derivatives, continue to be valid.
Under certain growth conditions on the second-order partial derivatives that
allow for explosive behavior near the boundaries, the almost sure rate in
Stute's representation of the empirical copula process can be recovered. The
conditions are verified, for instance, in the case of the Gaussian copula with
full-rank correlation matrix, many Archimedean copulas, and many extreme-value
copulas.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ387 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Max-stable models for multivariate extremes
Multivariate extreme-value analysis is concerned with the extremes in a
multivariate random sample, that is, points of which at least some components
have exceptionally large values. Mathematical theory suggests the use of
max-stable models for univariate and multivariate extremes. A comprehensive
account is given of the various ways in which max-stable models are described.
Furthermore, a construction device is proposed for generating parametric
families of max-stable distributions. Although the device is not new, its role
as a model generator seems not yet to have been fully exploited.Comment: Invited paper for RevStat Statistical Journal. 22 pages, 3 figure
On the covariance of the asymptotic empirical copula process
Conditions are given under which the empirical copula process associated with
a random sample from a bivariate continuous distribution has a smaller
asymptotic covariance function than the standard empirical process based on
observations from the copula. Illustrations are provided and consequences for
inference are outlined.Comment: 14 pages, 2 figure
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
Nonparametric estimation of multivariate extreme-value copulas
Extreme-value copulas arise in the asymptotic theory for componentwise maxima
of independent random samples. An extreme-value copula is determined by its
Pickands dependence function, which is a function on the unit simplex subject
to certain shape constraints that arise from an integral transform of an
underlying measure called spectral measure. Multivariate extensions are
provided of certain rank-based nonparametric estimators of the Pickands
dependence function. The shape constraint that the estimator should itself be a
Pickands dependence function is enforced by replacing an initial estimator by
its best least-squares approximation in the set of Pickands dependence
functions having a discrete spectral measure supported on a sufficiently fine
grid. Weak convergence of the standardized estimators is demonstrated and the
finite-sample performance of the estimators is investigated by means of a
simulation experiment.Comment: 26 pages; submitted; Universit\'e catholique de Louvain, Institut de
statistique, biostatistique et sciences actuarielle
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