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A geometric formulation of fiducial probability
- Publication Year :
- 2012
-
Abstract
- The geometric formulation of fiducial probability employed in this paper is an improvement over the usual pivotal quantity formulation. For a single parameter and single observation, the new formulation is based on the geometric properties of an ordinary two variable function and its surface representation. The following theorem is proved: A fiducial distribution for the continuous parameter $\theta$ exists if and only if (i) the continuous random probability distributions of $x$ for different $\theta$'s are non-intersecting, and (ii) the random distributions are complete, i.e. at the extreme values of $\theta$ the limiting probability distributions are zero and one for all $x$. The proof yields also a complete characterization of random distributions that lead to fiducial distributions. The paper also treats intersecting distributions and non-intersecting incomplete distributions. The latter, which are frequently encountered in a null hypothesis, are shown to be associated with intersecting "composite" distributions. An appendix compares the pivotal and geometric formulations.
- Subjects :
- Mathematics - Statistics Theory
62A99, 62F99
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1208.6578
- Document Type :
- Working Paper