The DOI matches its Crossref record; 40 of the 42 references it cites were found in a registry or an index.
The Trio Identity for Quasi-Monte Carlo Error
Open accessAbstract
Monte Carlo methods approximate integrals by sample averages of integrand values. The error of Monte Carlo methods may be expressed as a trio identity: the product of the variation of the integrand, the discrepancy of the sampling measure, and the confounding. The trio identity has different versions, depending on whether the integrand is deterministic or Bayesian and whether the sampling measure is deterministic or random. Although the variation and the discrepancy are common in the literature, the confounding is relatively unknown and under-appreciated. Theory and examples are used to show how the cubature error may be reduced by employing the low discrepancy sampling that defines quasi-Monte Carlo methods. The error may also be reduced by rewriting the integral in terms of a different integrand. Finally, the confounding explains why the cubature error might decay at a rate different from that of the discrepancy.
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“Such a measure plays a key role because if the confounding is high, that is, the error in approximation ˆ ν -ν is larger at locations where g tends to be larger (in magnitude), then we should expect a higher MCQMC error. Hence CNF (g, ˆ ν -ν) plays the same role as the data defect correlation ρR,G .”
From Statistical paradises and paradoxes in big data (I): Law of large populations, big data paradox, and the 2016 US presidential election (2018) · cites this paper as Hickernell (2018)
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“Indeed, pioneering work by Hickernell (2018) provides exactly that, and it generalizes the finite-sample version (5.2) to general (super-population) function forms.”
From Statistical paradises and paradoxes in big data (I): Law of large populations, big data paradox, and the 2016 US presidential election (2018) · cites this paper as Hickernell (2018)
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Cited by
Meng, X.-L. (2018) · The Annals of Applied Statistics · doi:10.1214/18-aoas1161sf · Cited by 316
Parton distributions need representative sampling
Courtoy, A., Huston, J. W., Nadolsky, P., Xie, K., Yan, M., & Yuan, C. –. (2023) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.107.034008 · Cited by 47
Fast automatic Bayesian cubature using lattice sampling
Jagadeeswaran, R., & Hickernell, F. J. (2019) · Statistics and Computing · doi:10.1007/s11222-019-09895-9 · Cited by 36
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Analysis of parton distributions in a pion with Bézier parametrizations
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Is a Transformed Low Discrepancy Design Also Low Discrepancy?
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Self-citationA Deterministic Sampling Method via Maximum Mean Discrepancy Flow with Adaptive Kernel
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From Data Defects to Response Functions: A View From Particle Physics
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Fast Automatic Bayesian Cubature Using Matching Kernels and DesignsPreprint
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Pronko, N. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1707.08658
Metadata
- Authors
- Hickernell, F. J.
- Affiliations
- Illinois Institute of Technology (US)
- Corresponding author
- Hickernell, Fred J.
- ORCID
- Hickernell, Fred J.
- Venue
- Springer Proceedings in Mathematics & Statistics
- Publisher
- Springer International Publishing
- Published
- 2018
- Pages
- 3-27
- ISSN
- 2194-1009 (ISSN-L)
- 2194-1017
- OpenAlex
- W2749558669
- Cited by
- 9
- References
- 42
- Citations by year
- 2026: 1 · 2024: 1 · 2023: 3 · 2021: 1 · 2020: 1 · 2019: 1 · 2018: 1
- FWCI
- 2.00
- Type
- conference paper
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Springer proceedings in mathematics & statistics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation (1522687)
- Topics
- Physical Sciences › Mathematics › Numerical Analysis › Mathematical Approximation and Integration
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical functions and polynomials
- Physical Sciences › Mathematics › Statistics and Probability › Markov Chains and Monte Carlo Methods
- Keywords
- quasi-Monte Carlo, discrepancy, confounding, low-discrepancy sampling, Bayesian integration
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