Covering numbers, dyadic chaining and discrepancy

Aistleitner, Christoph

doi:10.1016/j.jco.2011.03.001

Cited by 55 publications

(91 citation statements)

References 10 publications

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“…. , d given by 1) and puts N points in such a way that in every slice there is exactly one point. It is worth mentioning that for d = 1 Latin hypercube sampling is exactly the same as RSJ rank-1 lattice (namely simple stratified sampling).…”

Section: Negative Dependence Conditional Nqd Property and Pairwise mentioning

confidence: 99%

3. On negatively dependent sampling schemes, variance reduction, and probabilistic upper discrepancy bounds

Gnewuch¹,

Wnuk²,

Hebbinghaus³

2020

Discrepancy Theory

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We study some notions of negative dependence of a sampling scheme that can be used to derive variance bounds for the corresponding estimator or discrepancy bounds for the underlying random point set that are at least as good as the corresponding bounds for plain Monte Carlo sampling.We provide new pre-asymptotic bounds with explicit constants for the star discrepancy and the weighted star discrepancy of sampling schemes that satisfy suitable negative dependence properties. Furthermore, we compare the different notions of negative dependence and give several examples of negatively dependent sampling schemes, including mixed sequences.

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Section: Negative Dependence Conditional Nqd Property and Pairwise mentioning

confidence: 99%

3. On negatively dependent sampling schemes, variance reduction, and probabilistic upper discrepancy bounds

Gnewuch¹,

Wnuk²,

Hebbinghaus³

2020

Discrepancy Theory

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“…In fact the bound s 1=2 N 1=2 might be crucial: it is known that for all N 1 and s 1 there exists an N -element sequence having discrepancy Ä 10s 1=2 N 1=2 , but it is unknown how far this upper bound is from optimality. For more information we refer to [1], [10] and [12]. Remark 3.4] showed that in every bound of the form…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…We use a refined version of the dyadic partitioning technique in [1]. Let N 1, " > 0, Á > 0 and a parameter 10 be given ( will be chosen as a function of Á, see equation (3.20) below).…”

Section: Preliminariesmentioning

confidence: 99%

Probabilistic error bounds for the discrepancy of mixed sequences

Aistleitner

Hofer

2012

Monte Carlo Methods and Applications

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In many applications Monte Carlo (MC) sequences or Quasi-Monte Carlo (QMC) sequences are used for numerical integration. In moderate dimensions the QMC method typically yield better results, but its performance significantly falls off in quality if the dimension increases. One class of randomized QMC sequences, which try to combine the advantages of MC and QMC, are so-called mixed sequences, which are constructed by concatenating a d -dimensional QMC sequence and an (s d )-dimensional MC sequence to obtain a sequence in dimension s. Ökten, Tuffin and Burago proved probabilistic asymptotic bounds for the discrepancy of mixed sequences, which were refined by Gnewuch. In this paper we use an interval partitioning technique to obtain improved probabilistic bounds for the discrepancy of mixed sequences. By comparing them with lower bounds we show that our results are almost optimal.

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“…For a version with an explicit constant in this inequality we refer to [1], for a lower bound for arbitrary sets to [36], and for a corresponding lower bound of the expectation of the star discrepancy of a random point set to [20]. A standard reference for empirical processes is [98].…”

Section: Vapnik-červonenkis Classes and Empirical Processesmentioning

confidence: 99%

“…Quasi-Monte Carlo (QMC) rules are quadrature rules which can be used to approximate integrals defined on the s-dimensional unit cube [0, 1] …”

Section: Introductionmentioning

confidence: 99%