Testing multivariate uniformity and its applications

Jia Juan Liang; Kai Tai Fang; Fred J. Hickernell; Runze Li

doi:10.1090/s0025-5718-00-01203-5

Testing multivariate uniformity and its applications

Jia Juan Liang^*, Kai Tai Fang, Fred J. Hickernell, Runze Li

^*Corresponding author for this work

Research output: Contribution to journal › Journal article › peer-review

35 Citations (Scopus)

Abstract

Some new statistics are proposed to test the uniformity of random samples in the multidimensional unit cube [0, 1]^d (d ≥ 2). These statistics are derived from number-theoretic or quasi-Monte Carlo methods for measuring the discrepancy of points in [0, 1]^d. Under the null hypothesis that the samples are independent and identically distributed with a uniform distribution in [0, 1]^d, we obtain some asymptotic properties of the new statistics. By Monte Carlo simulation, it is found that the finite-sample distributions of the new statistics are well approximated by the standard normal distribution, N(0, 1), or the chi-squared distribution, χ²(2). A power study is performed, and possible applications of the new statistics to testing general multivariate goodness-of-fit problems are discussed.

Original language	English
Pages (from-to)	337-355
Number of pages	19
Journal	Mathematics of Computation
Volume	70
Issue number	233
Early online date	17 Feb 2000
DOIs	https://doi.org/10.1090/s0025-5718-00-01203-5
Publication status	Published - Jan 2001
Externally published	Yes

Scopus Subject Areas

Algebra and Number Theory
Computational Mathematics
Applied Mathematics

User-Defined Keywords

Discrepancy
Goodness-of-fit
Quasi-Monte Carlo methods
Testing uniformity

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10.1090/s0025-5718-00-01203-5

Cite this

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title = "Testing multivariate uniformity and its applications",

abstract = "Some new statistics are proposed to test the uniformity of random samples in the multidimensional unit cube [0, 1]d (d ≥ 2). These statistics are derived from number-theoretic or quasi-Monte Carlo methods for measuring the discrepancy of points in [0, 1]d. Under the null hypothesis that the samples are independent and identically distributed with a uniform distribution in [0, 1]d, we obtain some asymptotic properties of the new statistics. By Monte Carlo simulation, it is found that the finite-sample distributions of the new statistics are well approximated by the standard normal distribution, N(0, 1), or the chi-squared distribution, χ2(2). A power study is performed, and possible applications of the new statistics to testing general multivariate goodness-of-fit problems are discussed.",

keywords = "Discrepancy, Goodness-of-fit, Quasi-Monte Carlo methods, Testing uniformity",

author = "Liang, {Jia Juan} and Fang, {Kai Tai} and Hickernell, {Fred J.} and Runze Li",

note = "This work was partially supported by a Hong Kong Research Grants Council grant RGC/97-98/47.",

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doi = "10.1090/s0025-5718-00-01203-5",

language = "English",

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T1 - Testing multivariate uniformity and its applications

AU - Liang, Jia Juan

AU - Fang, Kai Tai

AU - Hickernell, Fred J.

AU - Li, Runze

N1 - This work was partially supported by a Hong Kong Research Grants Council grant RGC/97-98/47.

PY - 2001/1

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N2 - Some new statistics are proposed to test the uniformity of random samples in the multidimensional unit cube [0, 1]d (d ≥ 2). These statistics are derived from number-theoretic or quasi-Monte Carlo methods for measuring the discrepancy of points in [0, 1]d. Under the null hypothesis that the samples are independent and identically distributed with a uniform distribution in [0, 1]d, we obtain some asymptotic properties of the new statistics. By Monte Carlo simulation, it is found that the finite-sample distributions of the new statistics are well approximated by the standard normal distribution, N(0, 1), or the chi-squared distribution, χ2(2). A power study is performed, and possible applications of the new statistics to testing general multivariate goodness-of-fit problems are discussed.

AB - Some new statistics are proposed to test the uniformity of random samples in the multidimensional unit cube [0, 1]d (d ≥ 2). These statistics are derived from number-theoretic or quasi-Monte Carlo methods for measuring the discrepancy of points in [0, 1]d. Under the null hypothesis that the samples are independent and identically distributed with a uniform distribution in [0, 1]d, we obtain some asymptotic properties of the new statistics. By Monte Carlo simulation, it is found that the finite-sample distributions of the new statistics are well approximated by the standard normal distribution, N(0, 1), or the chi-squared distribution, χ2(2). A power study is performed, and possible applications of the new statistics to testing general multivariate goodness-of-fit problems are discussed.

KW - Discrepancy

KW - Goodness-of-fit

KW - Quasi-Monte Carlo methods

KW - Testing uniformity

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Testing multivariate uniformity and its applications

Abstract

Scopus Subject Areas

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