Fiducial Generalized Confidence Intervals

Generalized Pivotal Quantities and Generalized Confidence Intervals have proved to be useful tools for making inferences in many practical problems. Although generalized confidence intervals are not guaranteed to have exact frequentist coverage, a number of published and unpublished simulation studies suggest that the coverage probabilities of such intervals are sufficiently close to their nominal value to be useful in practice. In this paper we single out a subclass of generalized pivotal quantities, which we refer to as fiducial generalized pivotal quantities, and show that, under some mild conditions, generalized confidence inter-vals constructed using fiducial generalized pivotal quantities have correct frequentist coverage, at least asymptotically. We describe three general approaches for constructing fiducial generalized pivotal quan-tities – a recipe based on invertible pivotal relationships, and two extensions of it – and demonstrate their usefulness by deriving some previously unknown generalized pivotal quantities and generalized confidence intervals. It is fair to say that nearly every published generalized confidence interval can be obtained by using one of these recipes. As an interesting by-product of our investigations we note that the subfamily of fiducial generalized pivots has a close connection with fiducial inference proposed by R. A. Fisher. This is why we refer to the proposed generalized pivots as Fiducial Generalized Pivotal Quantities. We demonstrate these concepts using several examples.