Self-consistent confidence sets and tests of composite hypotheses applicable to restricted parameters

David R. Bickel and Alexandre G. Patriota

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Frequentist methods, without the coherence guarantees of fully Bayesian methods, are known to yield self-contradictory inferences in certain settings. The framework introduced in this paper provides a simple adjustment to $p$ values and confidence sets to ensure the mutual consistency of all inferences without sacrificing frequentist validity. Based on a definition of the compatibility of a composite hypothesis with the observed data given any parameter restriction and on the requirement of self-consistency, the adjustment leads to the possibility and necessity measures of possibility theory rather than to the posterior probability distributions of Bayesian and fiducial inference.

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Bernoulli, Volume 25, Number 1 (2019), 47-74.

Received: December 2014
Revised: March 2017
First available in Project Euclid: 12 December 2018

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bounded parameter deductive closure deductive cogency empty confidence set possibility theory $p$-value function ranking function ranking theory restricted parameter space surprise measure


Bickel, David R.; Patriota, Alexandre G. Self-consistent confidence sets and tests of composite hypotheses applicable to restricted parameters. Bernoulli 25 (2019), no. 1, 47--74. doi:10.3150/17-BEJ942.

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