Critical Values for Bivariate Student t-Tests

Abstract

Let t ₁ and t ₁₂ be two independent normally distributed random variables with zero means and equal variances, each divided by a common estimate of the standard deviations. The joint distribution of t ₁, t ₂ is a bivariate Student t-distribution. The integral of the joint frequency function over certain types of critical regions in the (t ₁, t ₂)-plane may be written as the linear combination of two definite integrals which are easy to compute. Critical values for five alternative hypotheses in simultaneous tests on t ₁ and t ₂ are presented, and approximations which are asymptotically exact are discussed.