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Biometrika 2002 89(2):359-374; doi:10.1093/biomet/89.2.359
© 2002 by Biometrika Trust
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Permutation tests for equality of distributions in high-dimensional settings

Peter Hall1 and Nader Tajvidi2

1 Centre for Mathematics and its Applications, Australian National University, Canberra 0200, Australia peter.hall@anu.edu.au 2 Centre for Mathematical Sciences, Lund Institute of Technology, Box 118, SE-221 00 Lund, Sweden nader@maths.lth.se

Motivated by applications in high-dimensional settings, we suggest a test of the hypothesis H0 that two sampled distributions are identical.It is assumed that two independent datasets are drawn from the respective populations, which may be very general. In particular, the distributions may be multivariate or infinite-dimensional, in the latter case representing, for example, the distributions of random functions from one Euclidean space to another. Our test uses a measure of distance between data. This measure should be symmetric but need not satisfy the triangle inequality, so it is not essential that it be a metric. The test is based on ranking the pooled dataset, with respect to the distance and relative to any fixed data value, and repeating this operation for each fixed datum. A permutation argument enables a critical point to be chosen such that the test has concisely known significance level, conditional on the set of all pairwise distances.

Key Words: Bootstrap; Functional data analysis; Hypergeometric distribution; Hypothesis test; Local alternative; Multivariate analysis; Rank test; Resampling

Received February 2001. Revised August 2001


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