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, [chi]2f, as a positive number n tends to infinity. Bartlett correction T was originally proposed so that its mean is coincident with the one of [chi]2f 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]2f 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 [chi]2f 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.