Generalized $h$-Statistics and Other Symmetric Functions

Tracy, D. S. ; Gupta, B. C.

Ann. Statist., Tome 2 (1974) no. 1, p. 837-844 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 1974-07-14
Classification:  62.10,  62.20,  Moment,  cumulant,  seminvariant,  $h$-statistic,  $k$-statistic,  generalized $h$-statistic,  polykay,  generalized $k$-statistic,  unbiased estimation,  finite population,  combinatorial coefficient,  symmetric mean,  power sum,  augmented monomial symmetric function,  power product sum,  unrestricted sum,  partition,  ordered partition