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

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.Canada Foundation for Innovation, the Ministry of Research and Innovation of Ontario, the Faculty of Medicine of the University of Ottaw