Edgeworth Coefficients for Standard Multivariate Estimates

I give for the first time explicit formulas for the coefficients needed for the fourth-order Edgeworth expansions of a multivariate standard estimate. I 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. I also give applications to ellipsoidal and hyperrectangular sets.