Generalized Confidence Intervals

Abstract

The purpose of this chapter is to develop a counterpart of generalized p-values in interval estimation. Even those practitioners who insist on conventional confidence intervals will find the generalization useful to obtain excellent approximate interval estimates for problems such the interval estimation in mixed models. According to simulation studies, such approximations have outperformed more complicated approximations reported in the literature. The conventional definition of a confidence interval will be generalized so that problems such as that of constructing exact interval estimates for the difference in two exponential means can be solved. As in the case of hypothesis testing, exact confidence intervals (classical) for statistical problems involving nuisance parameters are available only in special cases. For example, exact confidence intervals for the second moment of a normal distribution are not available. The Behrens-Fisher problem is probably the most well known problem in which exact confidence intervals based on minimal sufficient statistics do not exist. In Chapter 7 this problem will be discussed in detail and the problem will be solved by the generalized approach.