### 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, χ^{2}(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 (US) |
---|---|

Pages (from-to) | 337-355 |

Number of pages | 19 |

Journal | Mathematics of Computation |

Volume | 70 |

Issue number | 233 |

State | Published - Jan 2001 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Applied Mathematics
- Computational Mathematics

### Cite this

*Mathematics of Computation*,

*70*(233), 337-355.

}

*Mathematics of Computation*, vol. 70, no. 233, pp. 337-355.

**Testing multivariate uniformity and its applications.** / Liang, Jia Juan; Fang, Kai Tai; Hickernell, Fred J.; Li, Runze.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Testing multivariate uniformity and its applications

AU - Liang, Jia Juan

AU - Fang, Kai Tai

AU - Hickernell, Fred J.

AU - Li, Runze

PY - 2001/1

Y1 - 2001/1

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.

UR - http://www.scopus.com/inward/record.url?scp=0035626527&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035626527&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0035626527

VL - 70

SP - 337

EP - 355

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 233

ER -