Error Rates and Improved Algorithms for Rare Event Simulation with Heavy Weibull Tails

Asmussen, Søren; Kortschak, Dominik

doi:10.1007/s11009-013-9371-6

Cited by 14 publications

(9 citation statements)

References 21 publications

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“…These rates, or estimates for the rates, are obtained in some standard distribution classes. Their proofs are mainly based on sharp asymptotics of subexponential distributions obtained in [20][21][22]. Recall that some main classes of such distributions are RV(α), meaning regularly varying ones, where F(x) = L(x)/x α with α > 0 and L(·) are slowly varying, Weibull tails with F(x) = e −x α for some α ∈ (0, 1), and lognormal tails which are close to the case γ = 2 of F(x) = e − log γ x for x ≥ 1 and some γ > 1; we refer in the following to this class as lognormal type tails.…”

Section: Theoretical Resultsmentioning

confidence: 99%

Distinguishing Log-Concavity from Heavy Tails

Asmussen

Lehtomaa

2017

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Well-behaved densities are typically log-convex with heavy tails and log-concave with light ones. We discuss a benchmark for distinguishing between the two cases, based on the observation that large values of a sum X 1 + X 2 occur as result of a single big jump with heavy tails whereas X 1 , X 2 are of equal order of magnitude in the light-tailed case. The method is based on the ratio |X 1 − X 2 |/(X 1 + X 2 ), for which sharp asymptotic results are presented as well as a visual tool for distinguishing between the two cases. The study supplements modern non-parametric density estimation methods where log-concavity plays a main role, as well as heavy-tailed diagnostics such as the mean excess plot.

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Section: Theoretical Resultsmentioning

confidence: 99%

Distinguishing Log-Concavity from Heavy Tails

Asmussen

Lehtomaa

2017

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“…For instance, a study of some distributions having regularly varying hazard rate is given in Asmussen and Kortschak (2013). Interesting relations can be found between the class of subexponential distributions (denoted by S) and regularly varying hazard rates.…”

Section: Transformations Using Conversion Functionsmentioning

confidence: 99%

On tail dependence coefficients of transformed multivariate Archimedean copulas

Bernardino¹,

Rullière²

2016

Fuzzy Sets and Systems

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This paper presents the impact of a class of transformations of copulas in their upper and lower multivariate tail dependence coefficients. In particular we focus on multivariate Archimedean copulas. In the first part of this paper, we calculate multivariate tail dependence coefficients when the generator of the considered copula exhibits some regular variation properties, and we investigate the behaviour of these coefficients in cases that are close to tail independence. This first part exploits previous works of Charpentier and Segers (2009) and extends some results of Juri and Wüthrich (2003) and De Luca and Rivieccio (2012). In the second part of the paper we analyse the impact in the upper and lower multivariate tail dependence coefficients of a large class of transformations of dependence structures. These results are based on the transformations exploited by Di Bernardino and Rullière (2013a) and Di Bernardino and Rullière (2013b). We extend some bivariate results of Durante et al. (2010) in a multivariate setting by calculating multivariate tail dependence coefficients for transformed copulas. We obtain new results under specific conditions involving regularly varying hazard rates of components of the transformation. In the third part, we show the utility of using transformed Archimedean copulas, as they permit to build Archimedean generators exhibiting any chosen couple of lower and upper tail dependence coefficients. Furthermore, using results in Larsson and Nešlehová (2011), we detail the extreme behaviour of the transformed radial part of Archimedean copulas. At last, we explain possible applications with Markov chains with specific dependence structure.

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“…We finally give a brief survey of our results for the Weibull case F (x) = e −x β with 0 < β < 1 (related distributions, say modified by a power, are easily included, but for simplicity, we refrain from this). We refer to Asmussen and Kortschak (2012) for a more complete treatment. The density is f (x) = βx β−1 e −x β and f (x) = −p(x)F (x) where p(x) = β 2 x 2(β−1) + β(1 − β)x β−2 .…”

Section: The Weibull Casementioning

confidence: 99%

“…Remark 1). For subexponential distributions with a lighter tail than regular variation like the Weibull, a corresponding theory is developed in Asmussen and Kortschak (2012) and summarized in part in Section 5.…”

Section: Introductionmentioning

confidence: 99%