Efficient Simulation for Dependent Rare Events with Applications to Extremes

Andersen, Lars Nørvang; Laub, Patrick J.; Rojas-Nandayapa, Leonardo

doi:10.1007/s11009-017-9557-4

Cited by 3 publications

(2 citation statements)

References 40 publications

(58 reference statements)

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“…Sraj et al, 2014). More on detailed recent advances on EV copulae can be found in Gudendorf and Segers (2010), Kamnitui et al (2019), Vettori et al (2017), Andersen et al (2018), and Kim et al (2020).…”

Section: Bivariate Copulaementioning

confidence: 99%

Copulae: An overview and recent developments

Größer

Okhrin

2021

WIREs Computational Stats

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Over the decades that have passed since they were introduced, copulae still remain a very powerful tool for modeling and estimating multivariate distributions. This work gives an overview of copula theory and it also summarizes the latest results. This article recalls the basic definition, the most important cases of bivariate copulae, and it then proceeds to a sketch of how multivariate copulae are developed both from bivariate copulae and from scratch. Regarding higher dimensions, the focus is on hierarchical Archimedean, vine, and factor copulae, which are the most often used and most flexible ways to introduce copulae to multivariate distributions. We also provide an overview of how copulae can be used in various fields of data science, including recent results. These fields include but are not limited to time series and machine learning. Finally, we describe estimation and testing methods for copulae in general, their application to the presented copula structures, and we give some specific testing and estimation procedures for those specific copulae.

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Section: Bivariate Copulaementioning

confidence: 99%

Copulae: An overview and recent developments

Größer

Okhrin

2021

WIREs Computational Stats

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“…Another area where integrals like Eq. ( 1) are often encountered is in reliability analysis (Au & Beck, 2001a;Melchers & Beck, 2018;Andersen et al, 2018;Straub et al, 2020).…”

Section: Introductionmentioning

confidence: 99%

Integrals over Gaussians under Linear Domain Constraints

Gessner,

Kanjilal,

Hennig

2019

Preprint

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Integrals of linearly constrained multivariateGaussian densities are a frequent problem in machine learning and statistics, arising in tasks like generalized linear models and Bayesian optimization. Yet they are notoriously hard to compute, and to further complicate matters, the numerical values of such integrals may be very small. We present an efficient black-box algorithm that exploits geometry for the estimation of integrals over a small, truncated Gaussian volume, and to simulate therefrom. Our algorithm uses the Holmes-Diaconis-Ross (hdr) method combined with an analytic version of elliptical slice sampling (ess). Adapted to the linear setting, ess allows for efficient, rejection-free sampling, because intersections of ellipses and domain boundaries have closed-form solutions. The key idea of hdr is to decompose the integral into easier-to-compute conditional probabilities by using a sequence of nested domains. Remarkably, it allows for direct computation of the logarithm of the integral value and thus enables the computation of extremely small probability masses. We demonstrate the effectiveness of our tailored combination of hdr and ess on high-dimensional integrals and on entropy search for Bayesian optimization.

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