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CONFIDENCE INTERVALS UTILIZING PRIOR INFORMATION IN THE BEHRENS–FISHER PROBLEM
Author(s) -
Kabaila Paul,
Tuck Jarrod
Publication year - 2008
Publication title -
australian and new zealand journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 1369-1473
DOI - 10.1111/j.1467-842x.2008.00519.x
Subject(s) - mathematics , confidence interval , interval (graph theory) , combinatorics , coverage probability , tolerance interval , upper and lower bounds , distribution (mathematics) , statistics , class (philosophy) , cdf based nonparametric confidence interval , credible interval , degrees of freedom (physics and chemistry) , probability distribution , mathematical analysis , physics , quantum mechanics , artificial intelligence , computer science
Summary Consider two independent random samples of size f + 1 , one from an N (μ 1 , σ 2 1 ) distribution and the other from an N (μ 2 , σ 2 2 ) distribution, where σ 2 1 /σ 2 2 ∈ (0, ∞) . The Welch ‘approximate degrees of freedom’ (‘approximate t ‐solution’) confidence interval for μ 1 −μ 2 is commonly used when it cannot be guaranteed that σ 2 1 /σ 2 2 = 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 J 0 , has minimum coverage probability 1 −α. Now suppose that we have uncertain prior information that σ 2 1 /σ 2 2 = 1. We consider a broad class of confidence intervals for μ 1 −μ 2 with minimum coverage probability 1 −α. This class includes the interval J 0 , which we use as the standard against which other members of will be judged. A confidence interval utilizes the prior information substantially better than J 0 if (expected length of J )/(expected length of J 0 ) is (a) substantially less than 1 (less than 0.96, say) for σ 2 1 /σ 2 2 = 1 , and (b) not too much larger than 1 for all other values of σ 2 1 /σ 2 2 . 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 J 0 ) when σ 2 1 /σ 2 2 = 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 J 0 .