Improved statistics for contrasting means of two samples under non-normality

This paper presents the asymptotic expansions of the distributions of the two-sample t-statistic and the Welch statistic, for testing the equality of the means of two independent populations under non-normality. Unlike other approaches, we obtain the null distributions in terms of the distribution and density functions of the standard normal variable up to n^-1, where n is the pooled sample size. Based on these expansions, monotone transformations are employed to remove the higher-order cumulant effect. We show that the new statistics can improve the precision of statistical inference to the level of o (n^-1). Numerical studies are carried out to demonstrate the performance of the improved statistics. Some general rules for practitioners are also recommended.