Expansions for the distribution of asymptotically chi-square statistics

Suppose Xϵ→LNp(0,Σ) as ϵ→0ϵ→0 and Xϵ has a formal Edgeworth expansion in powers of ϵϵ. For example, Xϵ could be a standardized function of sample means of several independent random samples, with ϵ=n−1/2ϵ=n−1/2 and nn the minimum sample size.Let g be a function from RpRp to RqRq for which a linear transformation is available taking the moment generating function of any random variable X in RpRp to that of g(X). Then this can be used to compute the Edgeworth expansion for g(Xϵ).This approach is used to obtain a formal expansion for the distribution of |Xϵ|2 in terms of the chi-square distribution when Σ2=Σ. This case includes most ‘chi-square’ goodness-of-fit statistics as well as the standardized and Studentized statistics Xϵ′Σ−1Xϵ and Xϵ′Σ̂−1Xϵ for Σ positive-definite.