A Two-Dimensional T-Distribution and a New Test with Flexible Type I Error Control

Summary

The usual t-test for a null hypothesis $H_o = H_{o1} \wedge H_{o2} ,$,where $H_{oj} :\mu j = \mu _j^ * ,j = 1,2,$ concerning the means of two normal populations with equal but unknown variance, does not account for the possibility that one of the component hypotheses may be principal, i.e., scientifically more relevant or important than the other. In this we use some properties of a bivariate t-distribution to construct a test which permits asymmetric treatment of the two component hypotheses. The proposed test is shown to be unbiased and consistent.