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Methods of Tail Dependence Estimation
Characterization and quantification of climate extremes and their dependencies are fundamental to the studying of natural hazards. This chapter reviews various parametric and nonparametric tail dependence coefficient estimators. The tail dependence coefficient describes the dependence (degree of association) between concurrent extremes at different locations. Accurate and reliable knowledge of the spatial characteristics of extremes can help improve the existing methods of modeling the occurrence probabilities of extreme events. This chapter will review these methods and use two case studies to demonstrate the application of tail dependence analysis
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
An overview of the goodness-of-fit test problem for copulas
We review the main "omnibus procedures" for goodness-of-fit testing for
copulas: tests based on the empirical copula process, on probability integral
transformations, on Kendall's dependence function, etc, and some corresponding
reductions of dimension techniques. The problems of finding asymptotic
distribution-free test statistics and the calculation of reliable p-values are
discussed. Some particular cases, like convenient tests for time-dependent
copulas, for Archimedean or extreme-value copulas, etc, are dealt with.
Finally, the practical performances of the proposed approaches are briefly
summarized
Nonparametric estimation of extremal dependence
There is an increasing interest to understand the dependence structure of a
random vector not only in the center of its distribution but also in the tails.
Extreme-value theory tackles the problem of modelling the joint tail of a
multivariate distribution by modelling the marginal distributions and the
dependence structure separately. For estimating dependence at high levels, the
stable tail dependence function and the spectral measure are particularly
convenient. These objects also lie at the basis of nonparametric techniques for
modelling the dependence among extremes in the max-domain of attraction
setting. In case of asymptotic independence, this setting is inadequate, and
more refined tail dependence coefficients exist, serving, among others, to
discriminate between asymptotic dependence and independence. Throughout, the
methods are illustrated on financial data.Comment: 22 pages, 9 figure
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
Quotient correlation: A sample based alternative to Pearson's correlation
The quotient correlation is defined here as an alternative to Pearson's
correlation that is more intuitive and flexible in cases where the tail
behavior of data is important. It measures nonlinear dependence where the
regular correlation coefficient is generally not applicable. One of its most
useful features is a test statistic that has high power when testing nonlinear
dependence in cases where the Fisher's -transformation test may fail to
reach a right conclusion. Unlike most asymptotic test statistics, which are
either normal or , this test statistic has a limiting gamma
distribution (henceforth, the gamma test statistic). More than the common
usages of correlation, the quotient correlation can easily and intuitively be
adjusted to values at tails. This adjustment generates two new concepts--the
tail quotient correlation and the tail independence test statistics, which are
also gamma statistics. Due to the fact that there is no analogue of the
correlation coefficient in extreme value theory, and there does not exist an
efficient tail independence test statistic, these two new concepts may open up
a new field of study. In addition, an alternative to Spearman's rank
correlation, a rank based quotient correlation, is also defined. The advantages
of using these new concepts are illustrated with simulated data and a real data
analysis of internet traffic.Comment: Published in at http://dx.doi.org/10.1214/009053607000000866 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Copulas in finance and insurance
Copulas provide a potential useful modeling tool to represent the dependence structure
among variables and to generate joint distributions by combining given marginal
distributions. Simulations play a relevant role in finance and insurance. They are used to
replicate efficient frontiers or extremal values, to price options, to estimate joint risks, and so
on. Using copulas, it is easy to construct and simulate from multivariate distributions based
on almost any choice of marginals and any type of dependence structure. In this paper we
outline recent contributions of statistical modeling using copulas in finance and insurance.
We review issues related to the notion of copulas, copula families, copula-based dynamic and
static dependence structure, copulas and latent factor models and simulation of copulas.
Finally, we outline hot topics in copulas with a special focus on model selection and
goodness-of-fit testing
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
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
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