ON GENERALIZED FIDUCIAL INFERENCE

Abstract

In this paper we extend Fisher's fiducial argument and obtain a gener- alized fiducial recipe that greatly expands the applicability of fiducial ideas. We do this assuming as little structure as possible. We demonstrate this recipe on many examples of varying complexity. We investigate, by simulation and by theo- retical considerations, some properties of the statistical procedures derived by the generalized fiducial recipe observing very good performance. We also compare the properties of generalized fiducial inference to the properties of Bayesian inference. R. A. Fisher's fiducial inference has been the subject of many discussions and controversies ever since he introduced the idea during the 1930's. The idea experienced a bumpy ride, to say the least, during its early years and one can safely say that it eventually fell into disfavor among mainstream statisticians. However, it appears to have made a resurgence recently under the label of gen- eralized inference. In this new guise, fiducial inference has proved to be a useful tool for deriving statistical procedures for problems where frequentist methods with good properties were previously unavailable. Therefore we believe that the fiducial argument of R. A. Fisher deserves a fresh look from a new angle. Our main goal is to show that the idea of transferring randomness from the model to the parameter space seems to be a useful one—giving us a tool to design useful statistical methods. We depart from the usual tradition in several ways. When defining fiducial distribution we do not start with a pivotal quantity. In- stead we start with a data generating equation also called a structural equation. This often makes no difference to the final result but it gives us the added flex- ibility of being able to treat continuous and discrete data in a unified fashion. We then approach the definition of a fiducial probability as a simple transfer of probability measure. We then investigate some particular examples and notice that statistical methods designed using the fiducial reasoning have typically very ∗This is an exploratory paper that received special approval for exceeding the normal page limit set by