46 research outputs found
Bias correction in multivariate extremes
The estimation of the extremal dependence structure is spoiled by the impact
of the bias, which increases with the number of observations used for the
estimation. Already known in the univariate setting, the bias correction
procedure is studied in this paper under the multivariate framework. New
families of estimators of the stable tail dependence function are obtained.
They are asymptotically unbiased versions of the empirical estimator introduced
by Huang [Statistics of bivariate extremes (1992) Erasmus Univ.]. Since the new
estimators have a regular behavior with respect to the number of observations,
it is possible to deduce aggregated versions so that the choice of the
threshold is substantially simplified. An extensive simulation study is
provided as well as an application on real data.Comment: Published at http://dx.doi.org/10.1214/14-AOS1305 in the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Limit conditional distributions for bivariate vectors with polar representation
We investigate conditions for the existence of the limiting conditional
distribution of a bivariate random vector when one component becomes large. We
revisit the existing literature on the topic, and present some new sufficient
conditions. We concentrate on the case where the conditioning variable belongs
to the maximum domain of attraction of the Gumbel law, and we study geometric
conditions on the joint distribution of the vector. We show that these
conditions are of a local nature and imply asymptotic independence when both
variables belong to the domain of attraction of an extreme value distribution.
The new model we introduce can also be useful for simulations
Estimation of bivariate excess probabilities for elliptical models
Let be a random vector whose conditional excess probability
is of interest. Estimating this kind of
probability is a delicate problem as soon as tends to be large, since the
conditioning event becomes an extreme set. Assume that is elliptically
distributed, with a rapidly varying radial component. In this paper, three
statistical procedures are proposed to estimate for fixed ,
with large. They respectively make use of an approximation result of Abdous
et al. (cf. Canad. J. Statist. 33 (2005) 317--334, Theorem 1), a new second
order refinement of Abdous et al.'s Theorem 1, and a non-approximating method.
The estimation of the conditional quantile function
for large fixed is also addressed and these
methods are compared via simulations. An illustration in the financial context
is also given.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ140 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Dense classes of multivariate extreme value distributions
International audienceIn this paper, we explore tail dependence modelling in multivariate extreme value distributions. The measure of dependence chosen is the scale function, which allows combinations of distributions in a very flexible way. The correspondences between the scale function and the spectral measure or the stable tail dependence function are given. Combining scale functions by simple operations, three parametric classes of laws are (re)constructed and analyzed, and resulting nested and structured models are discussed. Finally, the denseness of each of these classes is shown
Clustered Archimax Copulas
When modeling multivariate phenomena, properly capturing the joint extremal
behavior is often one of the many concerns. Archimax copulas appear as
successful candidates in case of asymptotic dependence. In this paper, the
class of Archimax copulas is extended via their stochastic representation to a
clustered construction. These clustered Archimax copulas are characterized by a
partition of the random variables into groups linked by a radial copula; each
cluster is Archimax and therefore defined by its own Archimedean generator and
stable tail dependence function. The proposed extension allows for both
asymptotic dependence and independence between the clusters, a property which
is sought, for example, in applications in environmental sciences and finance.
The model also inherits from the ability of Archimax copulas to capture
dependence between variables at pre-extreme levels. The asymptotic behavior of
the model is established, leading to a rich class of stable tail dependence
functions.Comment: 42 pages, 10 figure
Slightly more births at full moon
A popular belief holds that the number of births highly increases when the moon is full. To test this belief, we use a 50-year data set of 38.7 million births in France. The signal includes quasi-periodic and discrete components that need to be subtracted. This is done using a non-linear Gaussian least-squares method. It results in residuals with very good statistical properties. A likelihood ratio test is used to reject that the residual means for the 30 days of the lunar month all equal 0 (p-value = ): the residuals show very small but highly significant variations in the lunar month due to an increase of births at full moon and the day after. The reason for the very small increase of birth at full moon is not investigated but can be suspected to result from a self-fulfilling prophecy
Slightly more births at full moon
A popular belief holds that the number of births highly increases when the moon is full. To test this belief, we use a 50-year data set of 38.7 million births in France. The signal includes quasi-periodic and discrete components that need to be subtracted. This is done using a non-linear Gaussian least-squares method. It results in residuals with very good statistical properties. A likelihood ratio test is used to reject that the residual means for the 30 days of the lunar month all equal 0 (p-value = ): the residuals show very small but highly significant variations in the lunar month due to an increase of births at full moon and the day after. The reason for the very small increase of birth at full moon is not investigated but can be suspected to result from a self-fulfilling prophecy
Estimation of conditional laws given an extreme component
Let be a bivariate random vector. The estimation of a probability of
the form is challenging when is large, and a
fruitful approach consists in studying, if it exists, the limiting conditional
distribution of the random vector , suitably normalized, given that
is large. There already exists a wide literature on bivariate models for which
this limiting distribution exists. In this paper, a statistical analysis of
this problem is done. Estimators of the limiting distribution (which is assumed
to exist) and the normalizing functions are provided, as well as an estimator
of the conditional quantile function when the conditioning event is extreme.
Consistency of the estimators is proved and a functional central limit theorem
for the estimator of the limiting distribution is obtained. The small sample
behavior of the estimator of the conditional quantile function is illustrated
through simulations.Comment: 32 pages, 5 figur
Generalized Logistic Models and its orthant tail dependence
The Multivariate Extreme Value distributions have shown their usefulness in
environmental studies, financial and insurance mathematics. The Logistic or
Gumbel-Hougaard distribution is one of the oldest multivariate extreme value
models and it has been extended to asymmetric models. In this paper we
introduce generalized logistic multivariate distributions. Our tools are
mixtures of copulas and stable mixing variables, extending approaches in Tawn
(1990), Joe and Hu (1996) and Foug\`eres et al. (2009). The parametric family
of multivariate extreme value distributions considered presents a flexible
dependence structure and we compute for it the multivariate tail dependence
coefficients considered in Li (2009)