On the Relative Approximation Error of the Generalized Pareto Approximation for a High Quantile

Beirlant, Jan; Raoult, Jean-Pierre; Worms, Rym

doi:10.1007/s10687-004-4724-0

Cited by 4 publications

(11 citation statements)

References 10 publications

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“…Before commenting the asymptotic behavior of ε ET (p n ; α n ), let us compare our results with [4].…”

Section: Application To the Et Approximationmentioning

confidence: 96%

“…In this paper, we focus on the asymptotic behavior of the extrapolation error (4). Indeed, in view of (2), the ET method extrapolates in the distribution tail from q(α n ) to q(p n ) thanks to an additive correction proportional to log(α n /p n ).…”

Section: Introductionmentioning

confidence: 99%

“…These results were extended to other maximum domains of attraction in [21,22] while penultimate approximations were established for the distribution of the excesses [27]. The relative extrapolation error induced by the approximation ofF un by the survival distribution function of a GPD is studied in [4].…”

Section: Introductionmentioning

confidence: 99%

“…Here, similarly to [4], we focus on the approximation of quantiles rather than approximations of distribution functions. Let us also highlight that these investigations are not limited to the ET method.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic behavior of the extrapolation error associated with the estimation of extreme quantiles

2020

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We investigate the asymptotic behavior of the (relative) extrapolation error associated with some estimators of extreme quantiles based on extreme-value theory. It is shown that the extrapolation error can be interpreted as the remainder of a first order Taylor expansion. Conditions are then provided such that this error tends to zero as the sample size increases. Interestingly, in case of the so-called Exponential Tail estimator, these conditions lead to a subdivision of Gumbel maximum domain of attraction into three subsets. In contrast, the extrapolation error associated with Weissman estimator has a common behavior over the whole Fréchet maximum domain of attraction. First order equivalents of the extrapolation error are then derived showing that Weissman estimator may lead to smaller extrapolation errors than the Exponential Tail estimator on some subsets of Gumbel maximum domain of attraction. The accuracy of the equivalents is illustrated numerically and an application on real data is also provided.

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“…Before commenting the asymptotic behavior of ε ET (p n ; α n ), let us compare our results with [4].…”

Section: Application To the Et Approximationmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Asymptotic behavior of the extrapolation error associated with the estimation of extreme quantiles

2020

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“…Similarly, the Pareto and Beta distribution functions correspond, respectively, to the Fré chet and the Weibull extreme value distribution functions. The generalized Pareto distributions were studied by many authors; see, for example, Davison and Smith (1990), Raoult and Worms (2003), Beirlant et al (2003), Rootzé n and Tajvidi (2006) and references therein. In this paper we investigate models for the asymptotic behavior of the excesses of M n over the high thresholds.…”

Section: Introductionmentioning

confidence: 99%

Coupon collector's problem and generalized Pareto distributions

Jocković

Mladenović

2011

Journal of Statistical Planning and Inference

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Estimation of bivariate excess probabilities for elliptical models

Abdous¹,

Fougères²,

Ghoudi³

et al. 2008

Bernoulli

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Let $(X,Y)$ be a random vector whose conditional excess probability $\theta(x,y):=P(Y\leq y | X>x)$ is of interest. Estimating this kind of probability is a delicate problem as soon as $x$ tends to be large, since the conditioning event becomes an extreme set. Assume that $(X,Y)$ is elliptically distributed, with a rapidly varying radial component. In this paper, three statistical procedures are proposed to estimate $\theta(x,y)$ for fixed $x,y$, with $x$ large. They respectively make use of an approximation result of Abdous et al. (cf. Canad. J. Statist. 33 (2005) 317--334, Theorem 1), a new second order refinement of Abdous et al.'s Theorem 1, and a non-approximating method. The estimation of the conditional quantile function $\theta(x,\cdot)^{\leftarrow}$ for large fixed $x$ is also addressed and these methods are compared via simulations. An illustration in the financial context is also given.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ140 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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On the Relative Approximation Error of the Generalized Pareto Approximation for a High Quantile

Cited by 4 publications

References 10 publications

Asymptotic behavior of the extrapolation error associated with the estimation of extreme quantiles

Asymptotic behavior of the extrapolation error associated with the estimation of extreme quantiles

Coupon collector's problem and generalized Pareto distributions

Estimation of bivariate excess probabilities for elliptical models

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