On integer partitions and the Wilcoxon rank-sum statistic

TL;DR

This paper presents a novel method using classical q-series to derive exact null distributions of the Wilcoxon rank-sum statistic, applicable even with tied data, avoiding normal approximation for small samples.

Contribution

It introduces a simple, algebraic approach for calculating exact distributions of the rank-sum statistic, extending previous methods to handle ties efficiently.

Findings

Method derived using q-series simplifies exact distribution calculations.

Applicable to data with ties, improving accuracy over normal approximation.

Potential for implementation in computer algebra systems for small sample analysis.

Abstract

In the literature, derivations of exact null distributions of rank-sum statistics is often avoided in cases where one or more ties exist in the data. By deriving the null distribution in the no-ties case with the aid of classical $q$-series results of Euler and Rothe, we demonstrate how a natural generalization of the method may be employed to derive exact null distributions even when one or more ties are present in the data. It is suggested that this method could be implemented in a computer algebra system, or even a more primitive computer language, so that the normal approximation need not be employed in the case of small sample sizes, when it is less likely to be very accurate. Several algorithms for determining exact distributions of the rank-sum statistic (possibly with ties) have been given in the literature (see Streitberg and R\"ohmel (1986) and Marx et al. (2016)), but none seem as simple as the procedure discussed here which amounts to multiplying out a certain polynomial, extracting coefficients, and finally dividing by a binomal coefficient.