Improved additive adjustments for the LR/ELR test statistics

The Bartlett adjustment, being a simple adjustment through division by the expected value of the test statistic, is commonly used as a general statistical tool to reduce the error of the chi-squared approximation of parametric/empirical likelihood ratio (LR/ELR) test statistic. In this paper, some improved test statistics in the additive forms are presented, whose errors of the chi-squared approximation are o(N−1), as in the case of the traditional multiplicative Bartlett adjustment, where N is the sample size. By deriving the N−1-difference of the power functions of two tests under a sequence of local alternatives, it is shown that none of several adjustments of the LR/ELR test statistic is uniformly superior. The results are numerically illustrated on specific examples.