Koksma–Hlawka Inequality

Fred J. Hickernell

doi:10.1002/0471667196.ess4085.pub2

Koksma–Hlawka Inequality

Fred J. Hickernell

Research output: Chapter in book/report/conference proceeding › Entry for encyclopedia/dictionary › peer-review

Abstract

The Koksma–Hlawka inequality is a tight error bound on the approximation of an integral by the sample average of integrand values. The integration error is bounded by a product of two terms, the discrepancy of the sample points, [equation image], and the variation of the integrand, V(g). These two quantities measure the quality of the sample points and the roughness of the integrand, respectively. The Koksma–Hlawka inequality plays a key role in the development of quasi–Monte Carlo methods. Such methods replace simple random sample points by low discrepancy points. The Koksma–Hlawka inequality has also influenced the study of experimental design and led to the creation of UNIFORM DESIGNS.

Original language	English
Title of host publication	Encyclopedia of Statistical Sciences
Editors	Samuel Kotz, Campbell B. Read, N. Balakrishnan, Brani Vidakovic, Norman L. Johnson
Number of pages	5
ISBN (Electronic)	9780471667193
DOIs	https://doi.org/10.1002/0471667196.ess4085.pub2
Publication status	Published - 15 Jul 2004
Externally published	Yes

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abstract = "The Koksma–Hlawka inequality is a tight error bound on the approximation of an integral by the sample average of integrand values. The integration error is bounded by a product of two terms, the discrepancy of the sample points, [equation image], and the variation of the integrand, V(g). These two quantities measure the quality of the sample points and the roughness of the integrand, respectively. The Koksma–Hlawka inequality plays a key role in the development of quasi–Monte Carlo methods. Such methods replace simple random sample points by low discrepancy points. The Koksma–Hlawka inequality has also influenced the study of experimental design and led to the creation of UNIFORM DESIGNS.",

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N2 - The Koksma–Hlawka inequality is a tight error bound on the approximation of an integral by the sample average of integrand values. The integration error is bounded by a product of two terms, the discrepancy of the sample points, [equation image], and the variation of the integrand, V(g). These two quantities measure the quality of the sample points and the roughness of the integrand, respectively. The Koksma–Hlawka inequality plays a key role in the development of quasi–Monte Carlo methods. Such methods replace simple random sample points by low discrepancy points. The Koksma–Hlawka inequality has also influenced the study of experimental design and led to the creation of UNIFORM DESIGNS.

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ER -

Koksma–Hlawka Inequality

Abstract

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