Pivotal Quantities for Wishart’s and Related Distributions, and a Paradox in Fiducial Theory

Summary An infinity of sets of pivotal quantities (with a remarkably simple distribution) is found for Wishart’s distribution, leading to an infinity of different “fiducial distributions” for the parameters, all derived from the same data. §7 is entirely independent of the rest of the paper, and provides the simplest case of the paradox discussed here. Pivotal quantities for the distribution of the second-order moments of a sample from a multivariate normal population are also found in the cases when the second-order moments are taken about the population mean and/or the sample is too small for the usual theory to apply.