Edgeworth Coefficients for Standard Multivariate Estimates

We give for the first time explicit formulas for the coefficients needed for the 4th order Edgeworth expansions of a multivariate standard estimate. We call these the Edgeworth coefficients. They are Bell polynomials in the cumulant coefficients. Standard estimates include most estimates of interest, including smooth functions of sample means and other empirical estimates. We also give applications to ellipsoidal and hyperrectangular sets.