Summaries of the papers listed hereLinking the HoeffdingSobol and Möbius formulas through a decomposition of Kuo, Sloan, Wasilkowski, and Wozniakowski
HoeffdingSobol decomposition of homogeneous cosurvival functions: from Choquet representation to extreme value theory application
The tail dependographAll characterizations of nondegenerate multivariate tail dependence structures are both functional and infinitedimensional. Taking advantage of the Hoeffding—Sobol decomposition, we derive new indices to measure and summarize the strength of dependence in a multivariate extreme value analysis. The tail superset importance coefficients provide a pairwise ordering of the asymptotic dependence structure. We then define the tail dependograph, which visually ranks the extremal dependence between the components of the random vector of interest. For the purpose of inference, a rankbased statistic is derived and its asymptotic behavior is stated. These new concepts are illustrated with both theoretical models and real data, showing that our methodology performs well in practice.Modeling extreme rainfall: A comparative study of spatial extreme value models
A standardized distancebased index to assess the quality of spacefilling designs
Adapting extreme value statistics to financial time series: dealing with bias and serial dependenceWe handle two major issues in applying extreme value analysis to financial time series, bias and serial dependence, jointly. This is achieved by studying bias correction method when observations exhibit weakly serial dependence, namely the \(\beta\)mixing series. For estimating the extreme value index, we propose an asymptotically unbiased estimator and prove its asymptotic normality under the \(\beta\)mixing condition. The bias correction procedure and the dependence structure have a joint impact on the asymptotic variance of the estimator. Then, we construct an asymptotically unbiased estimator of high quantiles. We apply the new method to estimate the ValueatRisk of the daily return on the Dow Jones Industrial Average Index. Bias correction in multivariate extremes
Environmental data: multivariate Extreme Value Theory in practice
Dense classes of multivariate extreme value distributionsIn 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 exible 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. Optimal rates of convergence in the Weibull model based on kerneltype estimators
Risk measures and multivariate extensions of Breiman’s theoremModeling insurance risks is a task that received an increasing attention because of Solvency Capital Requirements. The ruin probability has become a standard risk measure to assess regulatory capital. In this paper we focus on discrete time models for finite time horizon. Several results are available in the literature allowing to calibrate the ruin probability by means of the sum of the tail probabilities of individual claim amounts. The aim of this work is to obtain asymptotics for such probabilities under multivariate regularly variation and, more precisely, to derive them from Breiman’s Theorem extensions. We thus exhibit new situations where the ruin probability admits computable equivalents. Consequences are also derived in terms of the ValueatRisk. Semiparametric estimation for heavy tailed distributionsIn this paper, we generalize several works in the extreme value theory for the estimation of the extreme value index and the second order parameter. Weak consistency and asymptotic normality are proven under classical assumptions. Some numerical simulations and computations are also performed to illustrate the finitesample and the limiting behavior of the estimators. GNSS Integrity Achievement by using Extreme Value theoryThe demonstration of the GNSS integrity requirement (\(10^{7}\)/150 sec range) for SBAS services as for future systems like GALILEO is a key issue either in the development/acceptance phase or in the operational one. Currently, for SBAS as EGNOS or WAAS, a lot of simulations coupled with data collections were done before the operation or service commissioning and a permanent data collection network is used to monitor, among other parameters, the integrity (or more precisely the absence of a Loss Of Integrity (LOI) in case of misleading information). The demonstration needs to assess a \(10^{7}\) order of magnitude which is a tricky issue: the classical methods require several tens years of observations and such LOI are generally not observed among the data because of their scarcity. To make possible the extrapolation of the error distributions into the tails, CNES and TAS have established a research action with French Universities of Lyon I and Toulouse III to use the Extreme Value Theory (EVT). Recent developments in quantile estimation have allowed the application of EVT in numerous domains regardless of the underlying error distributions of the measurement data, and avoid the questionable assumption of Gaussian error distributions. … The likelihood ratio test for general mixture models with possibly structural parameter
This paper deals with the likelihood ratio test (LRT) for testing hypotheses on the mixing measure in mixture models with or without structural parameter. The main result gives the asymptotic distribution of the LRT statistics under some conditions that are proved to be almost necessary. A detailed solution is given for two testing problems: the test of a single distribution against any mixture, with application to Gaussian, Poisson and binomial distributions; the test of the number of populations in a finite mixture with or without structural parameter Numerical bounds for the distribution of the maximum of one and twodimensional processesWe consider the class of realvalued stochastic processes indexed on a compact subset of \(\mathbb{R}\) or \(\mathbb{R}^2\) with almost surely absolutely continuous sample paths. We obtain an implicit formula for the distributions of their maxima. The main result is the derivation of numerical bounds that turn out to be very accurate, in the Gaussian case, for levels that are not large. We also present the first explicit upper bound for the distribution tail of the maximum in the twodimensional Gaussian framework. Numerical comparisons are performed with known tools such as the Rice upper bound and expansions based on the Euler characteristic. We deal numerically with the determination of the persistence exponent. Asymptotic distribution and local power of the likelihood ratio test for mixturesWe consider the loglikelihood ratio test (LRT) for testing the number of components in a mixture of populations in a parametric family. We provide the asymptotic distribution of the LRT statistic under the null hypothesis as well as under contiguous alternatives when the parameter set is bounded. Moreover, for the simple contamination model we prove, under general assumptions, that the asymptotic local power under contiguous hypotheses may be arbitrarily close to the asymptotic level when the set of parameters is large enough. In the particular problem of normal distributions, we prove that, when the unknown mean is not a priori bounded, the asymptotic local power under contiguous hypotheses is equal to the asymptotic level. Asymptotic poisson character of extremes in nonstationary Gaussian modelsLet \(X\) be a nonstationary Gaussian process, asymptotically centered with constant variance. Let \(u\) be a positive real. Define \(R_u(t)\) as the number of upcrossings of level \(u\) by the process \(X\) on the interval \((0, t]\). Under some conditions we prove that the sequence of point processes \((R_u)_{u>0}\) converges weakly, after normalization, to a standard Poisson process as u tends to infinity. In consequence of this study we obtain the weak convergence of the normalized supremum to a Gumbel distribution.
