On the Equivalence of Polykays of the Second Degree and $\Sigma$'s

Abstract

A generalization of Tukey's polykays [3], [4], and [5], was made by Hooke [1] in reference to sampling from a two-way array or population. These generalized polykays were christened "bipolykays." Smith [2] also developed these functions of degree two independently. Working with certain structural patterns in the analysis of variance, a set of functions denoted by $\Sigma$'s were introduced by Wilk and Kempthorne [6] and formally defined and extended to include all "balanced" population structures by Zyskind [7]. Because of certain "nice" properties of symmetric means of degree two, the $\Sigma$ expansions were found to be relatively simple and could be defined for a large class of structures. In contrast, the work on the extension of the polykays was limited only to sampling from a two-way population structure though polykays of higher degrees were also considered. Zyskind [8] recognized the equivalence of Hooke's bipolykays of degree two and a certain subset of the $\Sigma$'s, and conjectured the equivalence of appropriately extended polykays with the whole set of $\Sigma$'s for all balanced structures. In this paper an extension of the bipolykays of degree two to "$n$-way-polykays" (henceforth referred to as generalized polykays) is made to encompass all balanced structures (as defined in [7]). (Since this paper was submitted, general definitions of polykays and symmetric means of all degrees have been formulated, properties of these developed, and the basic results applied to obtaining variances and co-variances of estimates of components of variation in certain two and three-factor balanced structures.) The equivalence of these generalized polykays and the $\Sigma$'s is then shown.