CONFIDENCE INTERVALS UTILIZING PRIOR INFORMATION IN THE BEHRENS–FISHER PROBLEM

Consider two independent random samples of size f + 1 , one from an N (μ₁, σ²₁) distribution and the other from an N (μ₂, σ²₂) distribution, where σ²₁/σ²₂∈ (0, ∞) . The Welch ‘approximate degrees of freedom’ (‘approximate t‐solution’) confidence interval for μ₁?μ₂ is commonly used when it cannot be guaranteed that σ²₁/σ²₂= 1 . Kabaila (2005, Comm. Statist. Theory and Methods 34 , 291–302) multiplied the half‐width of this interval by a positive constant so that the resulting interval, denoted by J₀, has minimum coverage probability 1 ?α. Now suppose that we have uncertain prior information that σ²₁/σ²₂= 1. We consider a broad class of confidence intervals for μ₁?μ₂ with minimum coverage probability 1 ?α. This class includes the interval J₀, which we use as the standard against which other members of will be judged. A confidence interval utilizes the prior information substantially better than J₀ if (expected length of J)/(expected length of J₀) is (a) substantially less than 1 (less than 0.96, say) for σ²₁/σ²₂= 1 , and (b) not too much larger than 1 for all other values of σ²₁/σ²₂ . For a given f, does there exist a confidence interval that satisfies these conditions? We focus on the question of whether condition (a) can be satisfied. For each given f, we compute a lower bound to the minimum over of (expected length of J)/(expected length of J₀) when σ²₁/σ²₂= 1 . For 1 ?α= 0.95 , this lower bound is not substantially less than 1. Thus, there does not exist any confidence interval belonging to that utilizes the prior information substantially better than J₀.