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The Behrens-Fisher Problem is a classical statistical problem. It is to testthe equality of the means of two normal populations using two independentsamples, when the equality of the population variances is unknown. Linnik(1968) has shown that this problem has no exact fixed-level tests based on thecomplete sufficient statistics. However, exact conventional solutions based onother statistics and approximate solutions based the complete sufficientstatistics do exist.  Existing methods are mainly asymptotic tests, and usually don't perform wellwhen the variances or sample sizes differ a lot. In this paper, we propose anew method to find an exact t-test (Te) to solve this classical Behrens-FisherProblem. Confidence intervals for the difference between two means areprovided. We also use detailed analysis to show that Te test reaches themaximum of degree of freedom and to give a weak version of proof that Te testhas the shortest confidence interval length expectation. Some simulations areperformed to show the advantages of our new proposed method compared toavailable conventional methods like Welch's test, paired t-test and so on. Wewill also compare it to unconventional method, like two-stage test.

Te Test: A New Non-asymptotic T-test for Behrens-Fisher Problems