Expansions for the distribution of asymptotically chi-square statistics

Suppose X-epsilon -> L N-p (0, Sigma) as epsilon -> 0 and X-epsilon has a formal Edgeworth expansion in powers of epsilon. For example, X-epsilon could be a standardized function of sample means of several independent random samples, with epsilon = n(-1/2) and n the minimum sample size. Let g be a function from R-p to R-q for which a linear transformation is available taking the moment generating function of any random variable X in R-p to that of g(X). Then this can be used to compute the Edgeworth expansion for g(X-epsilon). This approach is used to obtain a formal expansion for the distribution of vertical bar X-epsilon vertical bar(2) in terms of the chi-square distribution when Sigma(2) = Sigma. This case includes most 'chi-square' goodness-of-fit statistics as well as the standardized and Studentized statistics X'(epsilon)Sigma X--1(epsilon) and X'epsilon(Sigma) over cap X--1(epsilon) for Sigma positive-definite. (C) 2012 Elsevier B.V. All rights reserved.