The Behrens-Fisher Problem: A Z Distribution Approach
Ying-Ying Zhang *
Department of Statistics and Actuarial Science, College of Mathematics and Statistics, Chongqing University, Chongqing, China.
Teng-Zhong Rong
Department of Statistics and Actuarial Science, College of Mathematics and Statistics, Chongqing University, Chongqing, China.
Man-Man Li
Department of Statistics and Actuarial Science, College of Mathematics and Statistics, Chongqing University, Chongqing, China.
*Author to whom correspondence should be addressed.
Abstract
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.
Keywords: Behrens-Fisher problem, Z distribution, SIG distribution, two normal means, infinite series, exponential error structure.