The Annals of Statistics

The Small Sample Distribution of a Mann-Whitney Type Statistic for Circular Data

W. J. R. Eplett

Full-text: Open access


The union-intersection method of test construction can be used to derive from the Mann-Whitney rank test a test statistic for testing whether two samples of observations from circular distributions come from the same population. The null distribution of this test statistic will be investigated here. The approach followed is to use the principle of inclusion-exclusion to obtain an expression for the number of partitions of a positive integer which satisfy certain conditions. This enables the probabilities for values of the circular Mann-Whitney test statistic to be expressed explicitly in terms of the probabilities for values of the usual Mann-Whitney test statistic. Recurrence formulae enabling computation of the distribution for small sample sizes are given. There is a clear relationship between our work and results obtained by Steck for the Kolmogorov and Smirnov statistics. These results can also be derived from the present approach.

Article information

Ann. Statist., Volume 7, Number 2 (1979), 446-453.

First available in Project Euclid: 12 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G10: Hypothesis testing
Secondary: 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 05A17: Partitions of integers [See also 11P81, 11P82, 11P83]

Mann-Whitney rank test Smirnov statistics principle of inclusion-exclusion determinant generating function recurrence relation


Eplett, W. J. R. The Small Sample Distribution of a Mann-Whitney Type Statistic for Circular Data. Ann. Statist. 7 (1979), no. 2, 446--453. doi:10.1214/aos/1176344626.

Export citation