## The Annals of Statistics

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

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

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176344626

**Digital Object Identifier**

doi:10.1214/aos/1176344626

**Mathematical Reviews number (MathSciNet)**

MR520252

**Zentralblatt MATH identifier**

0414.62036

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62G10: Hypothesis testing

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

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aos/1176344626