Transformations with Improved Chi-Squared Approximations

Suppose that a nonnegative statistic T is asymptotically distributed as a chi-squared distribution with f degrees of freedom, χ2f, as a positive number n tends to infinity. Bartlett correction T was originally proposed so that its mean is coincident with the one of χ2f up to the order O(n−1). For log-likelihood ratio statistics, many authors have shown that the Bartlett corrections are asymptotically distributed as χ2f up to O(n−1), or with errors of terms of O(n−2). Bartlett-type corrections are an extension of Bartlett corrections to other statistics than log-likelihood ratio statistics. These corrections have been constructed by using their asymptotic expansions up to O(n−1). The purpose of the present paper is to propose some monotone transformations so that the first two moments of transformed statistics are coincident with the ones of χ2f 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 are 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 proposed transformations of these statistics give a larger improvement to the chi-squared approximation than do the Bartlett corrections. Further, it is seen that the proposed approximations are comparable with the approximation based on an Edgeworth expansion.