Fiducial distribution in a power series family

Abstract

In this article the interest is on finding the fiducial distribution of the parameter, when the probability distribution belongs to the power series family, as in Johnson et al. (1992 Johnson, N. L., S. Kotz, and A. W. Kemp. 1992. Univariate discrete distributions. New York, NY: John Wiley and Sons. Google Scholar]). Recently in Nájera and O’Reilly (2017 Nájera, E., and F. O’Reilly. 2017. On fiducial generators. Communications Statistics - Theory Methods 46 (5):2232–2248. doi:10.1080/03610926.2015.1040505.Taylor & Francis Online], Web of Science ®] , Google Scholar]) an argument is given to obtain a unique fiducial in the Bernoulli case. An attempt is made here to define some sort of invariance in a power series distribution so that, as was done in the Bernoulli case, one may find a unique invariant fiducial for the parameter. The Bernoulli case is reviewed in detail and the Poisson and negative binomial cases are addressed.