A New Procedure for the Behrens–Fisher Problem that Guarantees Confidence Levels

Abstract

This article considers the Behrens–Fisher problem of comparing two independent normal means without making any assumptions about the two unknown population variances. It is known that the natural statistic for this problem has a distribution that depends upon the ratio of the two variances, and that inferences can be made by using a t-distribution with degrees of freedom equal to the minimum of the degrees of freedom of the two variance estimates. This is the limiting distribution of the statistic as the variance ratio tends to zero or one. In this article a new procedure is developed that simultaneously provides inferences on both the means and the variance ratio. This procedure bounds the variance ratio away from zero and infinity, so it can allow inferences on the means that are sharper than those provided by the limiting distribution. Some numerical illustrations of the advantage provided by this new procedure are presented.

Keywords:

• Normal distribution
• Pairwise comparison
• Unequal variances
• Confidence interval
• Confidence set
• Test procedure
• p-Value
• Acceptance set
• Beta distribution
• Behrens–Fisher

Acknowledgments

I sincerely thank each of the reviewers for their helpful and insightful comments and suggestions that have resulted in a much improved version of this article.