CUMULANTS AND PARTITION LATTICES-II - GENERALIZED K-STATISTICS

Abstract

The role played by the Möbius function of the lattice of all partitions of a set in the theory of k-statistics and their generalisations is pointed out and the main results concerning these statistics are derived. The definitions and formulae for the expansion of products of generalised k-statistics are presented from this viewpoint and applied to arrays of random variables whose moments satisfy suitable symmetry constraints. Applications of the theory are given including the calculation of (joint) cumulants of k-statistics, the minimum variance estimation of (generalised) moments and the asymptotic behaviour of generalised k-statistics viewed as (reversed) martingales. © 1986, Australian Ma..