A non-recursive formula for various moments of the multivariate normal distribution with sectional truncation

Abstract

A unified formula for various moments of the multivariate normal distribution with sectional truncation is derived using a non-recursive method, where sectional truncation is given by several sections (regions) for selection including single and double truncation as special cases. The moments include raw, central, arbitrarily deviated, non-absolute, absolute and partially absolute moments with non-integer orders for variables taking absolute values. The formula is alternatively shown using weighted Kummer’s confluent hypergeometric function and, in the bivariate case, the weighted Gauss hypergeometric function, where the weighted functions have advantages of fast convergence. Numerical illustrations with simulations show that the methods employed are relatively free from accumulating cancellation errors.