Transformation tests and their asymptotic power in two-sample comparisons

The classical two-sample t-test is not robust to skewed populations, and the large sample approximation has low accuracy for finite sample sizes. This paper presents two new types of tests, the TCFU and the TT tests, for comparing populations with unequal variances. The TCFU test uses Welch's t-statistic as the test statistic and the Cornish-Fisher expansion as its critical values. The TT tests apply transformations to Welch's t-statistic and use the normal percentiles as critical values. Four monotone transformations are considered for the TT tests. We give asymptotic expansions for the power functions of the new tests accurate to the order of O(n(-1)). A comparison of different tests in terms of power and type I error is presented both theoretically and through Monte Carlo experiments. Analytical conditions are derived to help practitioners choose a powerful test.