Products of Two Polykays when One has Weight 5

Abstract

Fisher [5] introduced a combinatorial method to obtain sampling cumulants of $k$-statistics as linear functions of cumulants of an infinite parent population. Kendall [6] systematized Fisher's combinatorial method by providing rules for the same and their proofs. Tukey [11] considered the sample statistic $k_{rs\cdots}$ in order to simplify the presentation of sampling moment formulae of the $k$-statistics when samples are drawn from a finite population. These $k_{rs\cdots}$, termed generalized $k$-statistics by Abdel-Aty [1] and polykays$^2$ by Tukey [12], were in fact considered earlier by Dressel [3] for the seminvariant case $(r, s,\cdots \neq 1)$. Wishart [13] modified the combinatorial method to obtain products of $k$-statistics as linear combinations of polykays, obtained products of polykays by algebraic manipulation, and applied these to the case of a finite population. He provided formulae for products of $k$-statistics through weight 8, and of polykays through weight 6. These have appeared again in David, Kendall and Barton ([2], 196-200, Table 2.3). Schaeffer and Dwyer [8] provided formulae for products of seminvariant polykays through weight 8. Tracy [9] supplied formulae for all products of polykays of weight 7. Dwyer and Tracy [4] modified and extended the combinatorial method to obtain products of two polykays. They presented general formulae resulting from this method for products $k(P)k(Q)$, where $k(P) = k_P = k_{p1\cdots p\pi}$ is a polykay having any weight and weight $(Q) \leqq 4$. Such formulae may be looked upon as rules of multiplication of a polykay by another of weight up to 4. It is the purpose of this paper to extend these results to the case of weight $(Q) = 5$. The formulae are presented together for compactness in a tabular form in Table 1, each column of which reads a formula for some $Q$. Checks indicated in Section 4 are applied more easily in the tabular presentation. Illustrations showing the use of the formulae appear in Tables 2 and 3.