A modified score test statistic having chi-squared distribution to order n⁻¹

We give a general modified score test statistic whose nuil distribution is chi-squared to order n−1, where n is the sample size. The modified statistic depends on the joint cumulants of log likelihood derivatives for the full data. Some applications are discussed. Following Cox & Reid (1987a) we also derive a general formula for Bartlett-type corrections to improve test statistics whose null asymptotic distributions are chi-squared.