The Pareto Copula, Aggregation of Risks, and the Emperor's Socks

Klüppelberg, Claudia; Resnick, Sidney I.

doi:10.1017/s002190020000396x

Cited by 11 publications

(8 citation statements)

References 12 publications

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“…In analogy to standard copulas for random vectors we will employ a concept of a Lévy copula to capture the dependence structure within ν which dates back to Cont and Tankov (2004) and Kallsen and Tankov (2006). We will follow a slightly different approach due to Klüppelberg and Resnick (2008) and Eder and Klüppelberg (2012), however, and focus on nonparametric methods to assess the closely related Pareto-Lévy copula. Consult also Bollerslev, Todorov and Li (2013) for related work on jump dependence using extreme value theory.…”

mentioning

confidence: 99%

Nonparametric inference on Lévy measures and copulas

Bücher¹,

Vetter²

2013

Ann. Statist.

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In this paper nonparametric methods to assess the multivariate Lévy measure are introduced. Starting from high-frequency observations of a Lévy process X, we construct estimators for its tail integrals and the Pareto-Lévy copula and prove weak convergence of these estimators in certain function spaces. Given n observations of increments over intervals of length ∆n, the rate of convergence is k −1/2 n for kn = n∆n which is natural concerning inference on the Lévy measure. Besides extensions to nonequidistant sampling schemes analytic properties of the Pareto-Lévy 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.

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mentioning

confidence: 99%

Nonparametric inference on Lévy measures and copulas

Bücher¹,

Vetter²

2013

Ann. Statist.

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“…Then, the new sample can be writtenqs P x P I /x P (i) . The normalization to a Pareto scale is natural for establishing limit results in extreme-value theory (Klüppelberg and Resnick, 2008), but is much less used in "classical" statistical analyses. For describing further our resampling approach, we reformulate the above theoretical results by switching to a uniform scale X U = F (X) − 1 on the interval [−1, 0], which may lend itself to easier visual analysis and interpretation, especially when readers are familiar with the copula literature (Joe, 2014).…”

Section: Lifting Mechanismmentioning

confidence: 99%

Semi-parametric resampling with extremes

2021

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Nonparametric resampling methods such as Direct Sampling are powerful tools to simulate new datasets preserving important data features such as spatial patterns from observed or analogue datasets while using only minimal assumptions. However, such methods cannot generate extreme events beyond the observed range of data values. We here propose using tools from extreme value theory for stochastic processes to extrapolate observed data towards yet unobserved high quantiles. Original data are first enriched with new values in the tail region, and then classical resampling algorithms are applied to enriched data. In a first approach to enrichment that we label "naive resampling", we generate an independent sample of the marginal distribution while keeping the rank order of the observed data. We point out inaccuracies of this approach around the most extreme values, and therefore develop a second approach that works for datasets with many replicates. It is based on the asymptotic representation of extreme events through two stochastically independent components: a magnitude variable, and a profile field describing spatial variation. To generate enriched data, we fix a target range of return levels of the magnitude variable, and we resample magnitudes constrained to this range. We then use the second approach to generate heatwave scenarios of yet unobserved magnitude over France, based on daily temperature reanalysis training data for the years 2010 to 2016.

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“…then we write X ∈ RV (F \ C , a n , ν). For a normalized processes X * , obtained by standardizing marginals of X to unit Fréchet (e.g., Coles and Tawn, 1991, Section 5) or unit Pareto (Klüppelberg and Resnick, 2008), for instance, regular variation is equivalent to the convergence of the renormalised pointwise maximum n −1 max i=1,...,n X * i of independent replicates of X * to a nondegenerate process Z * , with unit Fréchet margins and exponent measure ν * (de Haan and Lin, 2001). The process Z * is called simple max-stable, and X * is said to lie in the max-domain of attraction of Z * .…”

Section: Functional Regular Variationmentioning

confidence: 99%

High-dimensional peaks-over-threshold inference

Fondeville

Davison

2018

Biometrika

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Max-stable processes are increasingly widely used for modelling complex extreme events, but existing fitting methods are computationally demanding, limiting applications to a few dozen variables. r-Pareto processes are mathematically simpler and have the potential advantage of incorporating all relevant extreme events, by generalizing the notion of a univariate exceedance. In this paper we investigate score matching for performing high-dimensional peaks over threshold inference, focusing on extreme value processes associated to log-Gaussian random functions and discuss the behaviour of the proposed estimators for regularly-varying distributions with normalized marginals. Their performance is assessed on grids with several hundred locations, simulating from both the true model and from its domain of attraction. We illustrate the potential and flexibility of our methods by modelling extreme rainfall on a grid with 3600 locations, based on risks for exceedances over local quantiles and for large spatially accumulated rainfall, and briefly discuss diagnostics of model fit. The differences between the two fitted models highlight the importance of the choice of risk and its impact on the dependence structure. * raphael.de-fondeville@epfl.ch † anthony.davison@epfl.ch 1 arXiv:1605.08558v2 [stat.ME]

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The Pareto Copula, Aggregation of Risks, and the Emperor's Socks

Cited by 11 publications

References 12 publications

Nonparametric inference on Lévy measures and copulas

Nonparametric inference on Lévy measures and copulas

Semi-parametric resampling with extremes

High-dimensional peaks-over-threshold inference

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