SYMMETRIC FUNCTIONS OF A TWO-WAY ARRAY'

摘要: 1. Summary. A family of polynomials in the elements a two-way array, or matrix, is introduced. This an extension, from sets to matrices, symmetric ki, k2, kil, k3, k12, etc., defined by Tukey [6], christened "polykays" [7], and which are generalization R. A. Fisher [1]. The present paper, called "bipolykays," functions sense that they invariant under permutation rows and/or columns matrix. paper defines bipolykays, shows inherited on average, develops formulas for use random pairing, provides tables conversion multiplication. description applications (see [2], [3], [41) will be postponed until later paper. These include (a) finding expressions sampling moments matrix "bisample" larger (b) (such as estimates variance components) associated with analysis table systematic interactions, (c) unbiased estimators variances covariances estimated components without interactions.