Improvement on Chi-Squared Approximation by Monotone Transformation

It is well known that Bartlett adjustment yields an improvement on the chi-squared approximations to the likelihood ratio test statistics. However, it is not possible to obtain such an improvement for some test statistics. Typical examples are the score test statistic and HotellingʹsT2-statistic. The purpose of the present paper is to give a general method for improving on the chi-squared approximation. The method suggested is based on a monotone transformation.