Distributional properties for the generalized p-value for the Behrens-Fisher problem

The generalized p-value method introduced by Tsui and Weerahandi [1989. Generalized p-values in significance testing of hypotheses in the presence of nuisance parameters. J. Amer. Statist. Assoc. 84 (406), 602-607] has been successfully used to provide small sample solutions for many hypothesis testing problems when nuisance parameters are present. Simulation studies show that generalized p-values have similar distributional properties as ordinary p-values. It is desirable to study theoretical properties of generalized p-values. Given a sample d, let p(d) be the generalized p-value for the Behrens-Fisher problem of testing the difference of two independent normal distribution means with possibly unequal distributional variances, as given in Tsui and Weerahandi [1989. Generalized p-values in significance testing of hypotheses in the presence of nuisance parameters. J. Amer. Statist. Assoc. 84 (406), 602-607]. We derive a closed form expression to show that, for small samples, the probability P(p(d)[less-than-or-equals, slant]r) is approximately less than or equal to r, for 0[less-than-or-equals, slant]r[less-than-or-equals, slant]0.5.