A Group Sequential Sampling Approach for the Behrens-Fisher Problem with Suspected Outliers in Data

In this paper, we take another look at the Behrens-Fisher Problem, which is a widely recognized problem in statistical literature. In fact, we work on a variant of the classical Behrens-Fisher problem setup, where we test whether the difference of two normal means equals a specified threshold value under the assumption of unknown and unequal variances. Under the proposed hypothesis testing framework, our aim is to control type I and type II error probabilities by imposing upper bounds on them. We establish that the resulting optimal fixed sample size depends on the unknown standard deviations, making the desired testing accuracies impossible to achieve. Therefore, we take a sequential hypothesis testing route and implement a group sequential sampling strategy to tackle the proposed testing problem. The associated termination rule (a random sample size used to estimate the optimal fixed sample size) is defined by using the improved classes of unbiased estimators for the population standard deviations instead of the customary sample standard deviations. This makes our sequential rule more robust under possible outlying observations in the data. We study several interesting properties of the proposed estimators and the subsequent sequential termination rule. Our group sequential sampling strategy proves to be first-order efficient and consistent. The extensive simulations and a real data illustration regarding diabetes patients further support our theoretical findings and highlight the practical relevance of the proposed methodology. Mathematics Subject Classifications: 62L12, 62F03, 62F05