Fiducial Inference - The Fiducial Distribution

The Fiducial Distribution

Fisher required the existence of a sufficient statistic for the fiducial method to apply. Suppose there is a single sufficient statistic for a single parameter. That is, suppose that the conditional distribution of the data given the statistic does not depend on the value of the parameter. For example suppose that n independent observations are uniformly distributed on the interval . The maximum, X, of the n observations is a sufficient statistic for ω. If only X is recorded and the values of the remaining observations are forgotten, these remaining observations are equally likely to have had any values in the interval . This statement does not depend on the value of ω. Then X contains all the available information about ω and the other observations could have given no further information.

The cumulative distribution function of X is

Probability statements about X/ω may be made. For example, given α, a value of a can be chosen with 0 < a < 1 such that

Thus

Then Fisher says that this statement may be inverted into the form

In this latter statement, ω is now regarded as a random variable and X is fixed, whereas previously it was the other way round. This distribution of ω is the fiducial distribution which may be used to form fiducial intervals.

The calculation is identical to the pivotal method for finding a confidence interval, but the interpretation is different. In fact older books use the terms confidence interval and fiducial interval interchangeably. Notice that the fiducial distribution is uniquely defined when a single sufficient statistic exists.

The pivotal method is based on a random variable that is a function of both the observations and the parameters but whose distribution does not depend on the parameter. Such random variables are called pivotal quantities. By using these, probability statements about the observations and parameters may be made in which the probabilities do not depend on the parameters and these may be inverted by solving for the parameters in much the same way as in the example above. However, this is only equivalent to the fiducial method if the pivotal quantity is uniquely defined based on a sufficient statistic.

A fiducial interval could be taken to be just a different name for a confidence interval and give it the fiducial interpretation. But the definition might not then be unique. Fisher would have denied that this interpretation is correct: for him, the fiducial distribution had to be defined uniquely and it had to use all the information in the sample.