373 research outputs found
Hybrid Copula Estimators
An extension of the empirical copula is considered by combining an estimator
of a multivariate cumulative distribution function with estimators of the
marginal cumulative distribution functions for marginal estimators that are not
necessarily equal to the margins of the joint estimator. Such a hybrid
estimator may be reasonable when there is additional information available for
some margins in the form of additional data or stronger modelling assumptions.
A functional central limit theorem is established and some examples are
developed.Comment: 17 page
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 Copulas
Being the limits of copulas of componentwise maxima in independent random
samples, extreme-value copulas can be considered to provide appropriate models
for the dependence structure between rare events. Extreme-value copulas not
only arise naturally in the domain of extreme-value theory, they can also be a
convenient choice to model general positive dependence structures. The aim of
this survey is to present the reader with the state-of-the-art in dependence
modeling via extreme-value copulas. Both probabilistic and statistical issues
are reviewed, in a nonparametric as well as a parametric context.Comment: 20 pages, 3 figures. Minor revision, typos corrected. To appear in F.
Durante, W. Haerdle, P. Jaworski, and T. Rychlik (editors) "Workshop on
Copula Theory and its Applications", Lecture Notes in Statistics --
Proceedings, Springer 201
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
Weak convergence of the weighted empirical beta copula process
The empirical copula has proved to be useful in the construction and
understanding of many statistical procedures related to dependence within
random vectors. The empirical beta copula is a smoothed version of the
empirical copula that enjoys better finite-sample properties. At the core lie
fundamental results on the weak convergence of the empirical copula and
empirical beta copula processes. Their scope of application can be increased by
considering weighted versions of these processes. In this paper we show weak
convergence for the weighted empirical beta copula process. The weak
convergence result for the weighted empirical beta copula process is stronger
than the one for the empirical copula and its use is more straightforward. The
simplicity of its application is illustrated for weighted Cram\'er--von Mises
tests for independence and for the estimation of the Pickands dependence
function of an extreme-value copula.Comment: 19 pages, 2 figure
Tails of correlation mixtures of elliptical copulas
Correlation mixtures of elliptical copulas arise when the correlation
parameter is driven itself by a latent random process. For such copulas, both
penultimate and asymptotic tail dependence are much larger than for ordinary
elliptical copulas with the same unconditional correlation. Furthermore, for
Gaussian and Student t-copulas, tail dependence at sub-asymptotic levels is
generally larger than in the limit, which can have serious consequences for
estimation and evaluation of extreme risk. Finally, although correlation
mixtures of Gaussian copulas inherit the property of asymptotic independence,
at the same time they fall in the newly defined category of near asymptotic
dependence. The consequences of these findings for modeling are assessed by
means of a simulation study and a case study involving financial time series.Comment: 21 pages, 3 figure
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