Quasi-random numbers for copula models

Cambou, Mathieu; Hofert, Marius; Lemieux, Christiane

doi:10.1007/s11222-016-9688-4

Cited by 22 publications

(17 citation statements)

References 37 publications

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“…, Y (d) may have a non-uniform marginal distribution, but rather that there may be a strong dependence in the joint distribution of these random variables (which by Sklar's theorem may be encoded in a so-called copula -see [26] for details). We refer to [12] for a discussion of these issues from a practitioner's point of view.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Tusnády’s Problem, the Transference Principle, and Non-uniform QMC Sampling

Aistleitner

Bilyk

Nikolov

2018

Springer Proceedings in Mathematics &Amp; Statistics

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It is well-known that for every N ≥ 1 and d ≥ 1 there exist point sets x1, . . . , xN ∈ [0, 1] d whose discrepancy with respect to the Lebesgue measure is of order at most (log N ) d−1 N −1 . In a more general setting, the first author proved together with Josef Dick that for any normalized measure µ on [0, 1] d there exist points x1, . . . , xN whose discrepancy with respect to µ is of order at most (log N ) (3d+1)/2 N −1 . The proof used methods from combinatorial mathematics, and in particular a result of Banaszczyk on balancings of vectors. In the present note we use a version of the so-called transference principle together with recent results on the discrepancy of red-blue colorings to show that for any µ there even exist points having discrepancy of order at most (log N ) d− 1 2 N −1 , which is almost as good as the discrepancy bound in the case of the Lebesgue measure.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Tusnády’s Problem, the Transference Principle, and Non-uniform QMC Sampling

Aistleitner

Bilyk

Nikolov

2018

Springer Proceedings in Mathematics &Amp; Statistics

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“…. , d. Since all E j are iid, there is a 1 d−k+1 probability that E k = E (1) . In such a case E k = x with probability 1 as we are given E (1) = x.…”

Section: Variance Analysis and Calibration Methodsmentioning

confidence: 99%

“…, n} by a low-discrepancy point set. The choice of sampling algorithm η is not very important to control the MC error, but it is for QMC, as explained in [1]. The sampling algorithms we propose in this work are applicable to both MC and QMC, and numerical results for both methods are reported in Section 6.…”

Section: Quasi-monte Carlo and Copula Modelsmentioning

confidence: 99%

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Importance Sampling and Stratification for Copula Models

Arbenz

Cambou²,

Hofert

et al. 2018

Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

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An importance sampling approach for sampling from copula models is introduced. The proposed algorithm improves Monte Carlo estimators when the functional of interest depends mainly on the behaviour of the underlying random vector when at least one of its components is large. Such problems often arise from dependence models in finance and insurance. The importance sampling framework we propose is particularly easy to implement for Archimedean copulas. We also show how the proposal distribution of our algorithm can be optimized by making a connection with stratified sampling. In a case study inspired by a typical insurance application, we obtain variance reduction factors sometimes larger than 1000 in comparison to standard Monte Carlo estimators when both importance sampling and quasi-Monte Carlo methods are used.

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“…Section 7 considers non-uniform transformations including importance sampling, Rosenblatt-Hlawka-Mück sequential inversion, and a non-uniform transformation on the unit simplex that yields the customary RQMC convergence rate for a class of functions including all polynomials on the simplex. While finishing up this paper we noticed that Cambou et al (2015) have also applied the Faa di Bruno formula in a QMC application, though they apply it to a different set of problems. They use it to give sufficient conditions for some integrands with respect to copulas to be in BVHK.…”

Section: Introductionmentioning

confidence: 99%

Transformations and Hardy--Krause Variation

Basu¹,

Owen²

2016

SIAM J. Numer. Anal.

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Using a multivariable Faa di Bruno formula we give conditions on transformations τ : [0, 1] m → X where X is a closed and bounded subset of R d such that f • τ is of bounded variation in the sense of Hardy and Krause for all f ∈ C d (X ). We give similar conditions for f •τ to be smooth enough for scrambled net sampling to attain O(n −3/2+ ) accuracy. Some popular symmetric transformations to the simplex and sphere are shown to satisfy neither condition. Some other transformations due to Fang and Wang (1993) satisfy the first but not the second condition. We provide transformations for the simplex that makes f • τ smooth enough to fully benefit from scrambled net sampling for all f in a class of generalized polynomials. We also find sufficient conditions for the Rosenblatt-HlawkaMück transformation in R 2 and for importance sampling to be of bounded variation in the sense of Hardy and Krause.

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Quasi-random numbers for copula models

Cited by 22 publications

References 37 publications

Tusnády’s Problem, the Transference Principle, and Non-uniform QMC Sampling

Tusnády’s Problem, the Transference Principle, and Non-uniform QMC Sampling

Importance Sampling and Stratification for Copula Models

Transformations and Hardy--Krause Variation

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