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## Impact of dependence on some multivariate risk indicators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/07/2015
Português

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The minimization of some multivariate risk indicators may be used as an
allocation method, as proposed in C\'enac et al. [6]. The aim of capital
allocation is to choose a point in a simplex, according to a given criterion.
In a previous paper [17] we proved that the proposed allocation technique
satisfies a set of coherence axioms. In the present one, we study the
properties and asymptotic behavior of the allocation for some distribution
models. We analyze also the impact of the dependence structure on the
allocation using some copulas.

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## Copula Mixture Model for Dependency-seeking Clustering

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/06/2012
Português

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We introduce a copula mixture model to perform dependency-seeking clustering
when co-occurring samples from different data sources are available. The model
takes advantage of the great flexibility offered by the copulas framework to
extend mixtures of Canonical Correlation Analysis to multivariate data with
arbitrary continuous marginal densities. We formulate our model as a
non-parametric Bayesian mixture, while providing efficient MCMC inference.
Experiments on synthetic and real data demonstrate that the increased
flexibility of the copula mixture significantly improves the clustering and the
interpretability of the results.; Comment: Appears in Proceedings of the 29th International Conference on
Machine Learning (ICML 2012)

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## Copula-based Hierarchical Aggregation of Correlated Risks. The behaviour of the diversification benefit in Gaussian and Lognormal Trees

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Quantitative Finance - Risk Management#Quantitative Finance - Computational Finance#Quantitative Finance - Portfolio Management#Quantitative Finance - Statistical Finance#91B30, 91B70, 62P05

The benefits of diversifying risks are difficult to estimate quantitatively
because of the uncertainties in the dependence structure between the risks.
Also, the modelling of multidimensional dependencies is a non-trivial task.
This paper focuses on one such technique for portfolio aggregation, namely the
aggregation of risks within trees, where dependencies are set at each step of
the aggregation with the help of some copulas. We define rigorously this
procedure and then study extensively the Gaussian Tree of quite arbitrary size
and shape, where individual risks are normal, and where the Gaussian copula is
used. We derive exact analytical results for the diversification benefit of the
Gaussian tree as a function of its shape and of the dependency parameters.
Such a "toy-model" of an aggregation tree enables one to understand the basic
phenomena's at play while aggregating risks in this way. In particular, it is
shown that, for a fixed number of individual risks, "thin" trees diversify
better than "fat" trees. Related to this, it is shown that hierarchical trees
have the natural tendency to lower the overall dependency with respect to the
dependency parameter chosen at each step of the aggregation. We also show that
these results hold in more general cases outside the gaussian world...

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## On Multivariate Extensions of Value-at-Risk

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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In this paper, we introduce two alternative extensions of the classical
univariate Value-at-Risk (VaR) in a multivariate setting. The two proposed
multivariate VaR are vector-valued measures with the same dimension as the
underlying risk portfolio. The lower-orthant VaR is constructed from level sets
of multivariate distribution functions whereas the upper-orthant VaR is
constructed from level sets of multivariate survival functions. Several
properties have been derived. In particular, we show that these risk measures
both satisfy the positive homogeneity and the translation invariance property.
Comparison between univariate risk measures and components of multivariate VaR
are provided. We also analyze how these measures are impacted by a change in
marginal distributions, by a change in dependence structure and by a change in
risk level. Illustrations are given in the class of Archimedean copulas.

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## Semiparametric Sparse Discriminant Analysis

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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In recent years, a considerable amount of work has been devoted to
generalizing linear discriminant analysis to overcome its incompetence for
high-dimensional classification (Witten & Tibshirani 2011, Cai & Liu 2011, Mai
et al. 2012, Fan et al. 2012). In this paper, we develop high-dimensional
semiparametric sparse discriminant analysis (HD-SeSDA) that generalizes the
normal-theory discriminant analysis in two ways: it relaxes the Gaussian
assumptions and can handle non-polynomial (NP) dimension classification
problems. If the underlying Bayes rule is sparse, HD-SeSDA can estimate the
Bayes rule and select the true features simultaneously with overwhelming
probability, as long as the logarithm of dimension grows slower than the cube
root of sample size. Simulated and real examples are used to demonstrate the
finite sample performance of HD-SeSDA. At the core of the theory is a new
exponential concentration bound for semiparametric Gaussian copulas, which is
of independent interest.; Comment: 34 pages, 1 figure

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## Spline approximations to conditional Archimedean copula

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/11/2013
Português

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We propose a flexible copula model to describe changes with a covariate in
the dependence structure of (conditionally exchangeable) random variables. The
starting point is a spline approximation to the generator of an Archimedean
copula. Changes in the dependence structure with a covariate $x$ are modelled
by flexible regression of the spline coefficients on $x$. The performances and
properties of the spline estimate of the reference generator and the abilities
of these conditional models to approximate conditional copulas are studied
through simulations. Inference is made using Bayesian arguments with posterior
distributions explored using importance sampling or adaptive MCMC algorithms.
The modelling strategy is illustrated with two examples.; Comment: Key words: Conditional copula ; Archimedean copula ; B-splines

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## Standardized drought indices: A novel uni- and multivariate approach

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/08/2015
Português

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As drought is among the natural hazards which affects people and economies
worldwide and often results in huge monetary losses sophisticated methods for
drought monitoring and decision making are needed. Several different approaches
to quantify drought have been developed during past decades. However, most of
these drought indices suffer from different shortcomings and do not account for
the multiple driving factors which promote drought conditions and their
inter-dependencies. We provide a novel methodology for the calculation of
(multivariate) drought indices, which combines the advantages of existing
approaches and omits their disadvantages. Moreover, our approach benefits from
the flexibility of vine copulas in modeling multivariate non-Gaussian
inter-variable dependence structures. A three-variate data example is used in
order to investigate drought conditions in Europe and to illustrate and reason
the different modeling steps. The data analysis shows the appropriateness of
the described methodology. Comparison to well-established drought indices shows
the benefits of our multivariate approach. The validity of the new methodology
is verified by comparing the spatial extent of historic drought events based on
different drought indices. Further...

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## Convex geometry of max-stable distributions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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It is shown that max-stable random vectors in $[0,\infty)^d$ with unit
Fr\'echet marginals are in one to one correspondence with convex sets $K$ in
$[0,\infty)^d$ called max-zonoids. The max-zonoids can be characterised as sets
obtained as limits of Minkowski sums of cross-polytopes or, alternatively, as
the selection expectation of a random cross-polytope whose distribution is
controlled by the spectral measure of the max-stable random vector.
Furthermore, the cumulative distribution function $\Prob{\xi\leq x}$ of a
max-stable random vector $\xi$ with unit Fr\'echet marginals is determined by
the norm of the inverse to $x$, where all possible norms are given by the
support functions of max-zonoids. As an application, geometrical
interpretations of a number of well-known concepts from the theory of
multivariate extreme values and copulas are provided. The convex geometry
approach makes it possible to introduce new operations with max-stable random
vectors.; Comment: 25 pages. Revised version

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## Quantile Spectral Processes: Asymptotic Analysis and Inference

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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Quantile- and copula-related spectral concepts recently have been considered
by various authors. Those spectra, in their most general form, provide a full
characterization of the copulas associated with the pairs $(X_t, X_{t-k})$ in a
process $(X_t)_{t \in \mathbb{Z}}$, and account for important dynamic features,
such as changes in the conditional shape (skewness, kurtosis),
time-irreversibility, or dependence in the extremes, that their traditional
counterpart cannot capture. Despite various proposals for estimation
strategies, only quite incomplete asymptotic distributional results are
available so far for the proposed estimators, which constitutes an important
obstacle for their practical application. In this paper, we provide a detailed
asymptotic analysis of a class of smoothed rank-based cross-periodograms
associated with the copula spectral density kernels introduced in Dette et al.
(2013). We show that, for a very general class of (possibly non-linear)
processes, properly scaled and centered smoothed versions of those
cross-periodograms, indexed by couples of quantile levels, converge weakly, as
stochastic processes, to Gaussian processes. A first application of those
results is the construction of asymptotic confidence intervals for copula
spectral density kernels. The same convergence results also provide asymptotic
distributions (under serially dependent observations) for a new class of
rank-based spectral methods involving the Fourier transforms of rank-based
serial statistics such as the Spearman...

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## A model-free characterization of recurrences in stationary time series

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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Study of recurrences in earthquakes, climate, financial time-series, etc. is
crucial to better forecast disasters and limit their consequences. However,
almost all the previous phenomenological studies involved only a long-ranged
autocorrelation function, or disregarded the multi-scaling properties induced
by potential higher order dependencies. Consequently, they missed the facts
that non-linear dependences do impact both the statistics and dynamics of
recurrence times, and that scaling arguments for the unconditional distribution
may not be applicable. We argue that copulas is the correct model-free
framework to study non-linear dependencies in time series and related concepts
like recurrences. Fitting and/or simulating the intertemporal distribution of
recurrence intervals is very much system specific, and cannot actually benefit
from universal features, in contrast to the previous claims. This has important
implications in epilepsy prognosis and financial risk management applications.; Comment: 4 pages, 2 figures, 2 proofs included in supplementary material

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## Distortion risk measures for sums of dependent losses

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Statistics - Methodology#Mathematics - Statistics Theory#Quantitative Finance - Risk Management#60B05, 62H20, 91B30

We discuss two distinct approaches, for distorting risk measures of sums of
dependent random variables, which preserve the property of coherence. The
first, based on distorted expectations, operates on the survival function of
the sum. The second, simultaneously applies the distortion on the survival
function of the sum and the dependence structure of risks, represented by
copulas. Our goal is to propose risk measures that take into account the
fluctuations of losses and possible correlations between risk components.; Comment: Accepted 25 October 2010, Journal Afrika Statistika Vol. 5, N9, 2010,
page 260--267

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## Nonparametric inference on L\'evy measures and copulas

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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In this paper nonparametric methods to assess the multivariate L\'{e}vy
measure are introduced. Starting from high-frequency observations of a L\'{e}vy
process $\mathbf{X}$, we construct estimators for its tail integrals and the
Pareto-L\'{e}vy copula and prove weak convergence of these estimators in
certain function spaces. Given n observations of increments over intervals of
length $\Delta_n$, the rate of convergence is $k_n^{-1/2}$ for $k_n=n\Delta_n$
which is natural concerning inference on the L\'{e}vy measure. Besides
extensions to nonequidistant sampling schemes analytic properties of the
Pareto-L\'{e}vy copula which, to the best of our knowledge, have not been
mentioned before in the literature are provided as well. We conclude with a
short simulation study on the performance of our estimators and apply them to
real data.; Comment: Published in at http://dx.doi.org/10.1214/13-AOS1116 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org)

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## Nonlinear filtering with correlated L\'evy noise characterized by copulas

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 19/08/2015
Português

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The objective in stochastic filtering is to reconstruct information about an
unobserved (random) process, called the signal process, given the current
available observations of a certain noisy transformation of that process.
Usually X and Y are modeled by stochastic differential equations driven by a
Brownian motion or a jump (or Levy) process.
We are interested in the situation where both the state process X and the
observation process Y are perturbed by coupled Levy processes. More precisely,
L=(L_1,L_2) is a 2--dimensional Levy process in which the structure of
dependence is described by a Levy copula. We derive the associated Zakai
equation for the density process and establish sufficient conditions depending
on the copula and $L$ for the solvability of the corresponding solution to the
Zakai equation. In particular, we give conditions of existence and uniqueness
of the density process, if one is interested to estimate quantities like P(
X(t)>a), where a is a threshold.; Comment: 32 pages

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## Semiparametric estimation of mutual information and related criteria : optimal test of independence

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 19/08/2015
Português

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#Mathematics - Statistics Theory#62F03, 62F05, 62F10, 62F12, 62F40, 62G05, 62G07, 62G09, 62G10,
62G20, 62G30

We derive independence tests by means of dependence measures thresholding in
a semiparametric context. Precisely, estimates of phi-mutual informations,
associated to phi-divergences between a joint distribution and the product
distribution of its margins, are derived through the dual representation of
phi-divergences. The asymptotic properties of the proposed estimates are
established, including consistency, asymptotic distributions and large
deviations principle. The obtained tests of independence are compared via their
relative asymptotic Bahadur efficiency and numerical simulations. It follows
that the proposed semiparametric Kullback-Leibler Mutual information test is
the optimal one. On the other hand, the proposed approach provides a new method
for estimating the Kullback-Leibler mutual information in a semiparametric
setting, as well as a model selection procedure in large class of dependency
models including semiparametric copulas.

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## An invitation to coupling and copulas: with applications to multisensory modeling

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/11/2015
Português

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#Statistics - Methodology#Quantitative Biology - Quantitative Methods#Quantitative Finance - Risk Management

This paper presents an introduction to the stochastic concepts of
\emph{coupling} and \emph{copula}. Coupling means the construction of a joint
distribution of two or more random variables that need not be defined on one
and the same probability space, whereas a copula is a function that joins a
multivariate distribution to its one-dimensional margins. Their role in
stochastic modeling is illustrated by examples from multisensory perception.
Pointers to more advanced and recent treatments are provided.

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## Statistical Modeling of Spatial Extremes

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/08/2012
Português

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The areal modeling of the extremes of a natural process such as rainfall or
temperature is important in environmental statistics; for example,
understanding extreme areal rainfall is crucial in flood protection. This
article reviews recent progress in the statistical modeling of spatial
extremes, starting with sketches of the necessary elements of extreme value
statistics and geostatistics. The main types of statistical models thus far
proposed, based on latent variables, on copulas and on spatial max-stable
processes, are described and then are compared by application to a data set on
rainfall in Switzerland. Whereas latent variable modeling allows a better fit
to marginal distributions, it fits the joint distributions of extremes poorly,
so appropriately-chosen copula or max-stable models seem essential for
successful spatial modeling of extremes.; Comment: Published in at http://dx.doi.org/10.1214/11-STS376 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org)

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## Generalized Logistic Models and its orthant tail dependence

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/04/2011
Português

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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).

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## VaR-Efficient Portfolios for a Class of Super- and Sub-Exponentially Decaying Assets Return Distributions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 06/01/2003
Português

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Using a family of modified Weibull distributions, encompassing both
sub-exponentials and super-exponentials, to parameterize the marginal
distributions of asset returns and their multivariate generalizations with
Gaussian copulas, we offer exact formulas for the tails of the distribution
$P(S)$ of returns $S$ of a portfolio of arbitrary composition of these assets.
We find that the tail of $P(S)$ is also asymptotically a modified Weibull
distribution with a characteristic scale $\chi$ function of the asset weights
with different functional forms depending on the super- or sub-exponential
behavior of the marginals and on the strength of the dependence between the
assets. We then treat in details the problem of risk minimization using the
Value-at-Risk and Expected-Shortfall which are shown to be (asymptotically)
equivalent in this framework.; Comment: Latex document of 33 pages including 1 table and 2 eps figures

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## Consistent testing for a constant copula under strong mixing based on the tapered block multiplier technique

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/06/2012
Português

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Considering multivariate strongly mixing time series, nonparametric tests for
a constant copula with specified or unspecified change point (candidate) are
derived; the tests are consistent against general alternatives. A tapered block
multiplier technique based on serially dependent multiplier random variables is
provided to estimate p-values of the test statistics. Size and power of the
tests in finite samples are evaluated with Monte Carlo simulations. The block
multiplier technique might have several other applications for statistical
inference on copulas of serially dependent data.; Comment: 34 pages

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## Sensitivity of the limit shape of sample clouds from meta densities

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

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The paper focuses on a class of light-tailed multivariate probability
distributions. These are obtained via a transformation of the margins from a
heavy-tailed original distribution. This class was introduced in Balkema et al.
(J. Multivariate Anal. 101 (2010) 1738-1754). As shown there, for the
light-tailed meta distribution the sample clouds, properly scaled, converge
onto a deterministic set. The shape of the limit set gives a good description
of the relation between extreme observations in different directions. This
paper investigates how sensitive the limit shape is to changes in the
underlying heavy-tailed distribution. Copulas fit in well with multivariate
extremes. By Galambos's theorem, existence of directional derivatives in the
upper endpoint of the copula is necessary and sufficient for convergence of the
multivariate extremes provided the marginal maxima converge. The copula of the
max-stable limit distribution does not depend on the margins. So margins seem
to play a subsidiary role in multivariate extremes. The theory and examples
presented in this paper cast a different light on the significance of margins.
For light-tailed meta distributions, the asymptotic behaviour is very sensitive
to perturbations of the underlying heavy-tailed original distribution...

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