Some statistical applications of Faa di Bruno

The formula of Faa di Bruno is used to calculate higher order derivatives of a composition of functions. In this paper, we first review the multivariate version due to Constantine and Savits [A multivariate Faa di Bruno formula with applications, Trans. AMS 348 (1996) 503-520]. We next derive some useful recursion formulas. These results are then applied to obtain both explicit expressions and recursive formulas for the multivariate Hermite polynomials and moments associated with a multivariate normal distribution. Finally, an explicit expression is derived for the formal Edgeworth series expansion of the distribution of a normalized sum of iid random variables.