Some Poset Statistics

Abstract

Statistics or functions are discussed that measure agreement between certain types of partially ordered data. These poset statistics are a generalization of two familiar classes of functions: the arrangement increasing functions and the decreasing reflection functions; those functions measure agreement between linearly ordered data. Specifically, the statistics in question are functions $h(\mathbf{X}_1, \mathbf{X}_2)$ of two matrix arguments, each having $N$ rows and they measure the agreement of the ordering of the $N$ rows of the two matrices. An example is used to illustrate and motivate the discussion. One statistic in this class is applied to the example; it generalizes Wilcoxon's rank sum statistic, Spearman's rank correlation and Page's statistic for ordered alternatives.