Fiducial Generalized Confidence Intervals

Generalized pivotal quantities (GPQs) and generalized confidence intervals (GCIs) have proven to be useful tools for making inferences in many practical problems. Although GCIs are not guaranteed to have exact frequentist coverage, a number of published and unpublished simulation studies suggest that the coverage probabilities of such intervals are sufficiently close to their nominal value so as to be useful in practice. In this article we single out a subclass of generalized pivotal quantities, which we call fiducial generalized pivotal quantities (FGPQs), and show that under some mild conditions, GCIs constructed using FGPQs have correct frequentist coverage, at least asymptotically. We describe three general approaches for constructing FGPQs-a recipe based on invertible pivotal relationships, and two extensions of it-and demonstrate their usefulness by deriving some previously unknown GPQs and GCIs. It is fair to say that nearly every published GCI can be obtained using one of these recipes. As an interesting byproduct 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.