Asymptotic expansions of moments and cumulants

Many parameters may be expanded as series with terms involving products of expectations and their estimates expanded as series involving products of averages. The computation of moments and cu-mulants of such estimates may be organized if the terms of the series expansions of parameters, estimates and their moments are considered as functions applied to lists. The lists form a vectors space associated with a quotient space of the polynomial algebra whose basis is the monomials involving random variables. Simple identities relating the functions induce transformations of lists. The computations reduce to the transformation of lists. The transformations themselves are computed for a collection of standard objects using a set of fundamental identities. These transformations are the bases for quite complex calculations. The framework permits the calculation of results which would otherwise involve complementary set partitions, k-statistics, and pattern functions. The examples given include the calculation of unbiased estimates of cumulants, of cumulants of these, and of moments of bootstrap estimates.