A function f([pi]) on the set of permutations of {1, 2, ..., n} is called arrangement increasing (AI) if it increases each time we transpose a pair of coordinates in descending order, i < j and [pi]i > [pi]j, putting them in ascending order. We define and develop a partial ordering <=AI on densities of rank vectors in terms of expectations of AI functions. Specially, one density g is defined to be AI-larger than another density f(f<=AI g) if the expectation under g of any AI function is at least as large as its expectation under f. We show that the uniform density is the AI-smallest AI density, and this leads to power results for tests of agreement of two rank vectors. The extreme points of the convex set of AI densities are determined, from which additional results concerning the minimum power of rank tests are shown to follow. We also give applications to ranking and selection problems.
