Best Attainable Rates of Convergence for Estimators of the Stable Tail Dependence Function

Drees, Holger; Huang, Xin

doi:10.1006/jmva.1997.1708

Cited by 109 publications

(108 citation statements)

References 17 publications

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“…, X nj . Replacing all distribution functions in (5.1) by their empirical counterparts and replacing t by k/n where k = k n is a positive sequence such that k n → ∞ and k n = o(n), we obtain the following nonparametric estimator for L n , called the empirical (stable) tail dependence function: Huang (1992), Drees and Huang (1998)]. The inclusion of the term 1/2 inside the indicators serves to improve the finite sample behavior of the estimator.…”

Section: 2mentioning

confidence: 99%

When uniform weak convergence fails: Empirical processes for dependence functions and residuals via epi- and hypographs

Bücher¹,

Segers²,

Volgushev³

2014

Ann. Statist.

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In the past decades, weak convergence theory for stochastic processes has become a standard tool for analyzing the asymptotic properties of various statistics. Routinely, weak convergence is considered in the space of bounded functions equipped with the supremum metric. However, there are cases when weak convergence in those spaces fails to hold. Examples include empirical copula and tail dependence processes and residual empirical processes in linear regression models in case the underlying distributions lack a certain degree of smoothness. To resolve the issue, a new metric for locally bounded functions is introduced and the corresponding weak convergence theory is developed. Convergence with respect to the new metric is related to epiand hypo-convergence and is weaker than uniform convergence. Still, for continuous limits, it is equivalent to locally uniform convergence, whereas under mild side conditions, it implies L p convergence. For the examples mentioned above, weak convergence with respect to the new metric is established in situations where it does not occur with respect to the supremum distance. The results are applied to obtain asymptotic properties of resampling procedures and goodness-of-fit tests.

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Section: 2mentioning

confidence: 99%

When uniform weak convergence fails: Empirical processes for dependence functions and residuals via epi- and hypographs

Bücher¹,

Segers²,

Volgushev³

2014

Ann. Statist.

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show abstract

“…The asymptotics of these estimators (or slight variants thereof) have been investigated in Huang (1992); Drees and Huang (1998); Schmidt and Stadtmüller (2006); Einmahl et al (2012); Bücher and Dette (2011), among others. In order to control the bias of the estimators one needs to assume a second order condition on the speed of convergence in the defining relation for Λ.…”

Section: Introductionmentioning

confidence: 99%

A note on nonparametric estimation of bivariate tail dependence

Bücher

2014

Statistics &Amp; Risk Modeling

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Nonparametric estimation of tail dependence can be based on a standardization of the marginals if their cumulative distribution functions are known. In this paper it is shown to be asymptotically more efficient if the additional knowledge of the marginals is ignored and estimators are based on ranks. The discrepancy between the two estimators is shown to be substantial for the popular Clayton model. A brief simulation study indicates that the asymptotic conclusions transfer to finite samples.

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“…, n} be such that k → ∞ and k/n → 0 as n → ∞. By replacing P by the empirical distribution function, t by k/n, and (2), we obtain the empirical tail dependence function (Huang, 1992;Drees and Huang, 1998) …”

Section: Estimating the Stable Tail Dependence Functionmentioning

confidence: 99%

Modelling Extremes Using Approximate Bayesian Computation

Erhardt¹,

Sisson²

2016

Extreme Value Modeling and Risk Analysis

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There is an increasing interest to understand the dependence structure of a random vector not only in the center of its distribution but also in the tails. Extreme-value theory tackles the problem of modelling the joint tail of a multivariate distribution by modelling the marginal distributions and the dependence structure separately. For estimating dependence at high levels, the stable tail dependence function and the spectral measure are particularly convenient. These objects also lie at the basis of nonparametric techniques for modelling the dependence among extremes in the max-domain of attraction setting. In case of asymptotic independence, this setting is inadequate, and more refined tail dependence coefficients exist, serving, among others, to discriminate between asymptotic dependence and independence. Throughout, the methods are illustrated on financial data.

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Best Attainable Rates of Convergence for Estimators of the Stable Tail Dependence Function

Cited by 109 publications

References 17 publications

When uniform weak convergence fails: Empirical processes for dependence functions and residuals via epi- and hypographs

When uniform weak convergence fails: Empirical processes for dependence functions and residuals via epi- and hypographs

A note on nonparametric estimation of bivariate tail dependence

Modelling Extremes Using Approximate Bayesian Computation

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