Statistical inference on difference or ratio of means from heteroscedastic normal populations

In this paper, we propose a two-stage sampling method (Stein, 1945) for making inferences on the means of two normal populations with heteroscedastic variances which is usually referred to as the Behrens–Fisher problem (Fisher, 1935 and Fisher, 1941). The two-stage method can be used to control Type-II error and the length of confidence interval conditional on a specified Type-I error or confidence level. Simulation results show that the two-stage method improves the powers of the tests and its performance exhibits superior robustness properties over the existing single-stage methods. Moreover, our method can be easily extended to compare multiple populations.