On Obtaining Permutation Distributions in Polynomial Time

Abstract

Abstract Polynomial time algorithms are presented for finding the permutation distribution of any statistic that is a linear combination of some function of either the original observations or the ranks. This class of statistics includes the original Fisher two-sample location statistic and such common nonparametric statistics as the Wilcoxon, Ansari-Bradley, Savage, and many others. The algorithms are presented for the two-sample problem and it is shown how to extend them to the multisample problem—for example, to find the distribution of the Kruskal-Wallis and other extensions of the Wilcoxon—and to the single-sample situation. Stratification, ties, and censored observations are also easily handled by the algorithms. The algorithms require polynomial time as opposed to complete enumeration algorithms, which require exponential time. This savings is effected by first calculating and then inverting the characteristic function of the statistic.