The Behrens-Fisher Problem: A Z Distribution Approach

We propose the Z distribution to tackle the Behrens-Fisher problem. First, we define the Z distribution which is a generalization of the t distribution, and then nd the pdf and cdf of the Z distribution. After that, we apply the Z distribution in the hypothesis testing of two normal means, where three different assumptions of the variances are considered. The Z distribution is very flexible in the applications in which one statistics that obeys the Z distribution is applicable to all the three assumptions of the variances. Finally, we provide two groups of simulation studies for the hypothesis testing problems of two normal means.