Fiducial consistency and group structure

Abstract

SUMMARY A consistency criterion is proposed for fiducial probability distributions: if a fiducial density function is used to modulate a prior density function, the resulting frequency function should be that obtained by a Bayesian analysis. For a real variable and a real parameter, the fulfilment of this criterion is equivalent to a condition examined by Lindley (1958) and it requires the parameter to be essentially a location parameter. For several variables and the same number of parameters, the criterion if applied to fiducial probability derived from a pivotal quantity produces a simple differential equation condition on the pivotal quantity. If one-dimensional fiducial distributions can be generated in an arbitrary direction from any point in the parameter space, then the fulfilment of the criterion requires the model to be of transformation-parameter form; for such models a variety of properties concerned with consistency and interpretation are available (Fraser, 1961). The fiducial method of inference was introduced by Fisher (1930) and has been discussed and developed in many of his subsequent papers. The development is primarily in terms of examples, with but little attention to precise governing principles. In this paper the method will be interpreted in terms of the pivotal quantities that are a common feature in the examples. A pivotal quantity is a function of a sufficient statistic and a parameter that has a fixed frequency distribution. For a single real variable and a single real parameter the distribution function is a pivotal quantity and it has the uniform distribution on (0, 1) regardless of the parameter value. In some statistical problems there may be a natural choice for the pivotal quantity based, perhaps on the physical origin of the problem. The quantity may then express a position, a probability or error position for an outcome relative to the parameter value. In other problems there may be a natural choice based on mathematical properties of the specification. In this paper a pivotal quantity will be assumed given as part of the statistical problem: the distribution of the pivotal variable will be interpreted as describing the error distribution or error pattern, and the pivotal equation as describing the way that the probability mass of the error pattern is applied to the sample space. The fiducial method as presented by Fisher involves a preliminary reduction to an exhaustive statistic. If the exhaustive statistic is of the same dimension as the parameter it is used as the basic variable in the pivotal quantity. If the exhaustive statistic is of higher dimension, then an ancillary statistic is needed such that conditionally the exhaustive statistic has the same dimension as the parameter and can be used as basic variable in the pivotal quantity.