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June, 1970 Products of Two Polykays when One has Weight 5
P. N. Nagambal, D. S. Tracy
Ann. Math. Statist. 41(3): 1114-1121 (June, 1970). DOI: 10.1214/aoms/1177696994

## Abstract

Fisher  introduced a combinatorial method to obtain sampling cumulants of $k$-statistics as linear functions of cumulants of an infinite parent population. Kendall  systematized Fisher's combinatorial method by providing rules for the same and their proofs. Tukey  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  and polykays$^2$ by Tukey , were in fact considered earlier by Dressel  for the seminvariant case $(r, s,\cdots \neq 1)$. Wishart  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 (, 196-200, Table 2.3). Schaeffer and Dwyer  provided formulae for products of seminvariant polykays through weight 8. Tracy  supplied formulae for all products of polykays of weight 7. Dwyer and Tracy  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.

## Citation

P. N. Nagambal. D. S. Tracy. "Products of Two Polykays when One has Weight 5." Ann. Math. Statist. 41 (3) 1114 - 1121, June, 1970. https://doi.org/10.1214/aoms/1177696994

## Information

Published: June, 1970
First available in Project Euclid: 27 April 2007

zbMATH: 0201.21302
Digital Object Identifier: 10.1214/aoms/1177696994  