A unifying framework for $k$-statistics, polykays and their multivariate generalizations

Abstract

Through the classical umbral calculus, we provide a unifying syntax for single and multivariate $k$-statistics, polykays and multivariate polykays. From a combinatorial point of view, we revisit the theory as exposed by Stuart and Ord, taking into account the Doubilet approach to symmetric functions. Moreover, by using exponential polynomials rather than set partitions, we provide a new formula for $k$-statistics that results in a very fast algorithm to generate such estimators.

Published in at http://dx.doi.org/10.3150/07-BEJ6163 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

Cited by in corpus (6)

• A new method for fast computing unbiased estimators of cumulants
• An instance of umbral methods in representation theory: the parking function module
• The classical umbral calculus: Sheffer sequences
• On the computation of classical, boolean and free cumulants
• Symbolic computation of moments of sampling distributions
• A new algorithm for computing the multivariate Faà di Bruno's formula