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July, 1974 Generalized $h$-Statistics and Other Symmetric Functions
D. S. Tracy, B. C. Gupta
Ann. Statist. 2(4): 837-844 (July, 1974). DOI: 10.1214/aos/1176342774


Dwyer's (1937) $h$-statistic is extended to the generalized $h$-statistic $h_{p_1\cdots p_u}$ such that $E(h_{p_1\cdots p_u}) = \mu_{p_1} \cdots \mu_{p_u}$, similar to the extension of Fisher's $k$-statistic to the generalized $k$-statistic $k_{p_1\cdots p_u}$ requiring $E(k_{p_1\cdots p_u}) = \kappa_{p_1} \cdots \kappa_{p_u}$. The $h$-statistics follow simpler multiplication rules than for $k$-statistics and involve smaller coefficients. Generalized $h$-statistics are studied in terms of symmetric means, unrestricted sums, and ordered partitions, and their relationships with generalized $k$-statistics are established. The statistics are useful in obtaining approximate forms for sampling distributions when parent population is not completely known.


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D. S. Tracy. B. C. Gupta. "Generalized $h$-Statistics and Other Symmetric Functions." Ann. Statist. 2 (4) 837 - 844, July, 1974.


Published: July, 1974
First available in Project Euclid: 12 April 2007

zbMATH: 0292.62016
MathSciNet: MR362631
Digital Object Identifier: 10.1214/aos/1176342774

Keywords: $h$-statistic , $k$-statistic , 62.10 , 62.20 , augmented monomial symmetric function , combinatorial coefficient , cumulant , finite population , generalized $h$-statistic , generalized $k$-statistic , Moment , ordered partition , Partition , Polykay , power product sum , power sum , seminvariant , symmetric mean , unbiased estimation , unrestricted sum

Rights: Copyright © 1974 Institute of Mathematical Statistics


Vol.2 • No. 4 • July, 1974
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