Chain rules for multivariate cumulant coefficients

Abstract

Cumulant coefficients are the building blocks of Edgeworth expansions and other analytic methods for statistical inference, such as bias reduction. Suppose that is a standard estimate of an unknown vector w and that t(·) is a smooth function on w. We show that is a standard estimate of t(w). We give its cumulant coefficients in terms of those of and the derivatives of t(w). These “chain rules” can then be used to get the cumulant coefficients of a smooth function of . The results are illustrated by two numerical examples.