×

Testing multivariate uniformity: the distance-to-boundary method. (English) Zbl 1115.62046

Summary: Given a random sample taken on a compact domain \({\mathcal S} \subset\mathbb{R}^d\), the authors propose a new method for testing the hypothesis of uniformity of the underlying distribution. The test statistic is based on the distance of every observation to the boundary of \({\mathcal S}\). The proposed test has a number of interesting properties. In particular, it is feasible and particularly suitable for high dimensional data; it is distribution free for a wide range of choices of \({\mathcal S}\); it can be adapted to the case that the support of \({\mathcal S}\) is unknown; and it also allows for one-sided versions.
Moreover, the results suggest that, in some cases, this procedure does not suffer from the well-known curse of dimensionality. The authors study the properties of this test from both a theoretical and practical point of view. In particular, an extensive Monte Carlo simulation study allows them to compare their methods with some alternative procedures. They conclude that the proposed test provides quite a satisfactory balance between power, computational simplicity, and adaptability to different dimensions and supports.

MSC:

62G10 Nonparametric hypothesis testing
62H15 Hypothesis testing in multivariate analysis
65C05 Monte Carlo methods

Software:

spatial
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Apostol, Figures circumscribing circles, American Mathematical Monthly 111 pp 853– (2004) · Zbl 1187.51012
[2] Baringhaus, Testing for spherical symmetry of a multivariate distribution, The Annals of Statistics 19 pp 899– (1991) · Zbl 0725.62053
[3] Cuevas, On boundary estimation, Advances in Applied Probability 36 pp 340– (2004) · Zbl 1045.62019
[4] Deheuvels, Strong bounds for multidimensional spacings, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 62 pp 465– (1983) · Zbl 0488.60041
[5] Devroye, Detection of abnormal behavior via nonparametric estimation of the support, SIAM Journal of Applied Mathematics 38 pp 480– (1980) · Zbl 0479.62028
[6] Diggle, Statistical Analysis of Spatial Point Patterns (2003)
[7] Hall, Geometric representation of high dimension, low sample size data, Journal of the Royal Statistical Society Series B 67 pp 427– (2005) · Zbl 1069.62097
[8] Janson, Maximal spacings in several dimensions, The Annals of Probability 15 pp 274– (1987) · Zbl 0626.60017
[9] Justel, A multivariate Kolmogorov-Smirnov test of goodness of fit, Statistics & Probability Letters 35 pp 251– (1997)
[10] Liang, Testing multivariate uniformity and its applications, Mathematics of Computation 70 pp 337– (2001) · Zbl 0958.65016
[11] Mardia, Measures of multivariate skewness and kurtosis with applications, Biometrika 57 pp 519– (1970) · Zbl 0214.46302
[12] Moller, Statistical Inference and Simulation for Spatial Point Processes (2004)
[13] Reschehofer, Length tests for goodness of fit, Biometrika 78 pp 207– (1991)
[14] Ripley, Tests of ’randomness’ for spatial point patterns, Journal of the Royal Statistical Society Series B 39 pp 172– (1979) · Zbl 0427.62065
[15] Rukhin, Testing randomness: a suite of statistical procedures, Theory of Probability and its Applications 45 pp 111– (2001)
[16] Silverman, Density Estimation for Statistics and Data Analysis (1986) · Zbl 0617.62042
[17] Székely, A new test for multivariate normality, Journal of Multivariate Analysis 93 pp 58– (2005) · Zbl 1087.62070
[18] Zuo, General notions of statistical depth function, The Annals of Statistics 28 pp 461– (2000) · Zbl 1106.62334
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.