Transformations with improved chi-squared approximations

Suppose that a nonnegative statistic T is asymptotically distributed as a c hi-squared distribution with f degrees of freedom, chi(f)(2), as a positive number n tends to infinity. Bartlett correction (T) over tilde was origina lly proposed so that its mean is coincident with the one of chi(f)(2) up to the order O(n(-1)). For log-likelihood ratio statistics, many authors have shown that the Bartlett corrections are asymptotically distributed as chi( f)(2) up to O(n(-1)), or with errors of terms of O(n(-2)). Bartlett-type co rrections are an extension of Bartlett corrections to other statistics than log-likelihood ratio statistics. These corrections have been constructed b y using their asymptotic expansions up to O(n(-1)). The purpose of the pres ent paper is to propose some monotone transformations so that the first two moments of transformed statistics are coincident with the ones of chi(f)(2 ) up to O(n(-1)). It may be noted that the proposed transformations can be applied to a wide class of statistics whether their asymptotic expansions a re available or not. A numerical study of some test statistics that are not a log-likelihood ratio statistic is discribed. It is shown that the propos ed transformations of these statistics give a larger improvement to the chi -squared approximation than do the Bartlett corrections. Further, it is see n that the proposed approximations are comparable with the approximation ba sed on an Edgeworth expansion. (C) 2000 Academic Press.