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## Impact of dependence on some multivariate risk indicators

Maume-Deschamps, Véronique; Rullière, Didier; Said, Khalil
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.41%
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.

## Copula Mixture Model for Dependency-seeking Clustering

Rey, Melanie; Roth, Volker
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.41%
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)

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

Bruneton, Jean-Philippe
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.41%
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...

## On Multivariate Extensions of Value-at-Risk

Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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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.

## Semiparametric Sparse Discriminant Analysis

Mai, Qing; Zou, Hui
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.41%
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

## Spline approximations to conditional Archimedean copula

Lambert, Philippe
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.41%
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

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

Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.41%
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...

## Convex geometry of max-stable distributions

Molchanov, Ilya
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.41%
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

## Quantile Spectral Processes: Asymptotic Analysis and Inference

Kley, Tobias; Volgushev, Stanislav; Dette, Holger; Hallin, Marc
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.41%
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...

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

Chicheportiche, Rémy; Chakraborti, Anirban
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.41%
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

## Distortion risk measures for sums of dependent losses

Brahimi, Brahim; Meraghni, Djamel; Necir, Abdelhakim
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.41%
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

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

Bücher, Axel; Vetter, Mathias
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.41%
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)

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

Fernando, B. P. W.; Hausenblas, E.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.41%
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

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

Keziou, Amor; Regnault, Philippe
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.41%
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.

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

Colonius, Hans
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.41%
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.

## Statistical Modeling of Spatial Extremes

Davison, A. C.; Padoan, S. A.; Ribatet, M.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.41%
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)

## Generalized Logistic Models and its orthant tail dependence

Ferreira, Helena; Pereira, Luísa
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.41%
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).

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

Malevergne, Y.; Sornette, D.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.41%
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

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

Bücher, Axel; Ruppert, Martin
Tipo: Artigo de Revista Científica