Comment: Fuzzy and Randomized Confidence Intervals and P-Values

Professor Geyer and Professor Meeden have given us an intriguing article with much material for thought and exploration, and they deserve our congratulations. Although the idea of randomized procedures has long existed, this paper has revitalized the discussion on randomized confidence intervals and randomized P -values. Interval estimation of a binomial proportion is a very basic but very important problem with an extensive literature. Brown, Cai and DasGupta (2001) revisited this problem and showed that the performance of the standard Wald interval, which is used extensively in textbooks and in practice, is far more erratic and inadequate than is appreciated. Several natural alternative confidence intervals for p were recommended in Brown, Cai and DasGupta (2001). See also Agresti and Coull (1998). These intervals are all what the authors call crisp intervals. The coverage probability of these crisp confidence intervals contains significant oscillation, which is intrinsic in all crisp intervals due to the lattice structure of the binomial distributions. In the present paper, Geyer and Meeden introduce the notion of fuzzy con- fidence intervals with the goal to eliminate oscillation and to have the exact coverage probability. The con- fidence intervals are obtained by inverting families of randomized tests. In addition, the authors introduce the notion of fuzzy P -values. The introduction of the critical function φ as a function of three variables x, α and θ provides a unified description of fuzzy decision, fuzzy confidence interval and fuzzy P -values. Our discussion here will focus on four issues: (1) What is new in this paper?; (2) exact versus approximate coverage; (3) expected length; (4) generalization of abstract randomized confidence intervals to simultaneous inference