Generalized $h$-Statistics and Other Symmetric Functions

Abstract

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.