New Methods for Multivariate Normal Moments

Multivariate normal moments are foundational for statistical methods. The derivation and simplification of these moments are critical for the accuracy of various statistical estimates and analyses. Normal moments are the building blocks of the Hermite polynomials, which in turn are the building blocks of the Edgeworth expansions for the distribution of parameter estimates. Isserlis (1918) gave the bivariate normal moments and two special cases of trivariate moments. Beyond that, convenient expressions for multivariate variate normal moments are still not available. We compare three methods for obtaining them, the most powerful being the differential method. We give simpler formulas for the bivariate moment than that of Isserlis, and explicit expressions for the general moments of dimensions 3 and 4.