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Coherent fiducial distributions

20 August 2013

D. R. Bickel and M. Padilla, “A prior-free framework of coherent inference and its derivation of simple shrinkage estimators,” Journal of Statistical Planning and Inference 145, 204–221 (2014). 2012 version

This paper proposes a new method of shrinking point and interval estimates on the basis of coherent fiducial inference. Since problems with the interpretation of fiducial probability have prevented its widespread use, this manuscript first places fiducial inference within a general framework that has Bayesian and frequentist foundations.

The certainty distribution is a non-frequentist concept arising from decision theories of subjective Bayesianism. Interpreting the confidence distribution as a certainty distribution results in a coherent fiducial distribution. In some ways, this framework is a development of Wilkinson’s 1977 confidence-based fiducial theory.

The table in the discussion section compares Bayesian and various non-Bayesian posterior distributions. The theorem in Section 2 explains why such distributions are practical for statistical inference even when they are not confidence distributions. The simulations quantify frequentist performance of point and interval shrinkage estimators based on coherent fiducial distributions.