Properties of Students $t$ and of the Behrens-Fisher Solution to the Two Means Problem

Abstract

Conditional properties of the usual confidence intervals for the situations referred to in the title are investigated. It is shown that there can be no negatively biased relevant selections in a sense which implies that there can be no negatively biased relevant subsets in the sense of Buehler (1959). The intuitive meaning of these results is that there is no way of betting that the quoted confidence levels are too high which yields positive expected return for all parameter values. In addition it is reported that the coverage probabilities for the Behrens-Fisher intervals are always larger than the nominal significance level would suggest. Thus the Behrens-Fisher and Student's $t$ procedures can be considered to be conservative.