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

David R. Bickel and Alexandre G. Patriota

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

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

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

Permanent link to this document

Digital Object Identifier

Zentralblatt MATH identifier

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.

Export citation


  • [1] Ball, F.G., Britton, T. and O’Neill, P.D. (2002). Empty confidence sets for epidemics, branching processes and Brownian motion. Biometrika 89 211–224.
  • [2] Berger, J.O. (2003). Could Fisher, Jeffreys and Neyman have agreed on testing? Statist. Sci. 18 1–32.
  • [3] Bickel, D.R. (2012). The strength of statistical evidence for composite hypotheses: Inference to the best explanation. Statist. Sinica 22 1147–1198.
  • [4] Bickel, D.R. (2013). Minimax-optimal strength of statistical evidence for a composite alternative hypothesis. Int. Stat. Rev. 81 188–206.
  • [5] Bickel, D.R. (2013). Pseudo-likelihood, explanatory power, and Bayes’s theorem [Comment on “A likelihood paradigm for clinical trials”] [MR3196591]. J. Stat. Theory Pract. 7 178–182.
  • [6] Bickel, D.R. and Padilla, M. (2014). A prior-free framework of coherent inference and its derivation of simple shrinkage estimators. J. Statist. Plann. Inference 145 204–221.
  • [7] Chuaqui, R. (1991). Truth, Possibility and Probability: New Logical Foundations of Probability and Statistical Inference. North-Holland Mathematics Studies 166. Amsterdam: North-Holland.
  • [8] Cohen, L. (1992). An Essay on Belief and Acceptance. Oxford: Clarendon Press.
  • [9] Cox, D.R. (1977). The role of significance tests. Scand. J. Stat. 4 49–70.
  • [10] De Baets, B., Tsiporkova, E. and Mesiar, R. (1999). Conditioning in possibility theory with strict order norms. Fuzzy Sets and Systems 106 221–229.
  • [11] Dubois, D., Foulloy, L., Mauris, G. and Prade, H. (2004). Probability-possibility transformations, triangular fuzzy sets, and probabilistic inequalities. Reliab. Comput. 10 273–297.
  • [12] Dubois, D., Moral, S. and Prade, H. (1997). A semantics for possibility theory based on likelihoods. J. Math. Anal. Appl. 205 359–380.
  • [13] Dubois, D. and Prade, H. (1998). Possibility theory: Qualitative and quantitative aspects. In Quantified Representation of Uncertainty and Imprecision. Handb. Defeasible Reason. Uncertain. Manag. Syst. 1 169–226. Dordrecht: Kluwer Academic.
  • [14] Edwards, A.W.F. (1992). Likelihood. Baltimore, MD: Johns Hopkins Univ. Press.
  • [15] Efron, B. and Tibshirani, R. (1998). The problem of regions. Ann. Statist. 26 1687–1718.
  • [16] Fisher, R.A. (1973). Statistical Methods and Scientific Inference. New York: Hafner Press.
  • [17] Fraser, D., Reid, N. and Wong, A. (2004). Inference for bounded parameters. Phys. Rev. D 69 033002.
  • [18] Fraser, D.A.S. (2011). Is Bayes posterior just quick and dirty confidence? Statist. Sci. 26 299–316.
  • [19] Gabriel, K.R. (1969). Simultaneous test procedures – Some theory of multiple comparisons. Ann. Math. Stat. 40 224–250.
  • [20] Ghasemi Hamed, M., Serrurier, M. and Durand, N. (2012). Representing uncertainty by possibility distributions encoding confidence bands, tolerance and prediction intervals. In Scalable Uncertainty Management. Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 7520 233–246.
  • [21] Giang, P.H. and Shenoy, P.P. (2005). Decision making on the sole basis of statistical likelihood. Artificial Intelligence 165 137–163.
  • [22] Hacking, I. (2001). An Introduction to Probability and Inductive Logic. Cambridge: Cambridge Univ. Press.
  • [23] Jeffreys, H. (1948). Theory of Probability. Oxford: Oxford Univ. Press.
  • [24] Kaplan, M. (1996). Decision Theory as Philosophy. Cambridge: Cambridge Univ. Press.
  • [25] Lapointe, S. and Bobée, B. (2000). Revision of possibility distributions: A Bayesian inference pattern. Fuzzy Sets and Systems 116 119–140.
  • [26] Lavine, M. and Schervish, M.J. (1999). Bayes factors: What they are and what they are not. Amer. Statist. 53 119–122.
  • [27] Mandelkern, M. (2002). Setting confidence intervals for bounded parameters. Statist. Sci. 17 149–172.
  • [28] Marchand, E. and Strawderman, W.E. (2004). Estimation in restricted parameter spaces: A review. In A Festschrift for Herman Rubin. Institute of Mathematical Statistics Lecture Notes – Monograph Series 45 21–44. Beachwood, OH: IMS.
  • [29] Marchand, É. and Strawderman, W.E. (2013). On Bayesian credible sets, restricted parameter spaces and frequentist coverage. Electron. J. Stat. 7 1419–1431.
  • [30] Marchioni, E. (2006). Possibilistic conditioning framed in fuzzy logics. Internat. J. Approx. Reason. 43 133–165.
  • [31] Masson, M.-H. and Denœux, T. (2006). Inferring a possibility distribution from empirical data. Fuzzy Sets and Systems 157 319–340.
  • [32] Mauris, G., Lasserre, V. and Foulloy, L. (2001). A fuzzy approach for the expression of uncertainty in measurement. Measurement 29 165–177.
  • [33] Nadarajah, S., Bityukov, S. and Krasnikov, N. (2015). Confidence distributions: A review. Stat. Methodol. 22 23–46.
  • [34] Patriota, A.G. (2013). A classical measure of evidence for general null hypotheses. Fuzzy Sets and Systems 233 74–88.
  • [35] Peirce, C.S. (1998). The Essential Peirce: Selected Philosophical Writings (18931913). The Essential Peirce: Selected Philosophical Writings. Bloomington: Indiana Univ. Press.
  • [36] Polansky, A.M. (2007). Observed Confidence Levels: Theory and Application. New York: Chapman and Hall.
  • [37] Puhalskii, A. (1997). Large deviations of semimartingales: A maxingale problem approach. I. Limits as solutions to a maxingale problem. Stoch. Stoch. Rep. 61 141–243.
  • [38] Puhalskii, A. (2001). Large Deviations and Idempotent Probability. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 119. Boca Raton, FL: Chapman & Hall/CRC.
  • [39] Royall, R.M. (1997). Statistical Evidence: A Likelihood Paradigm. Monographs on Statistics and Applied Probability 71. London: Chapman & Hall.
  • [40] Schervish, M.J. (1996). $P$ values: What they are and what they are not. Amer. Statist. 50 203–206.
  • [41] Schweder, T. and Hjort, N.L. (2002). Confidence and likelihood. Scand. J. Stat. 29 309–332.
  • [42] Shackle, G. (1961). Decision, Order and Time in Human Affairs. Cambridge: Cambridge Univ. Press.
  • [43] Silvapulle, M.J. and Sen, P.K. (2005). Constrained Statistical Inference: Order, Inequality, and Shape Constraints. Wiley Series in Probability and Statistics. New York: John Wiley & Sons.
  • [44] Spohn, W. (2012). The Laws of Belief: Ranking Theory and Its Philosophical Applications. Oxford: Oxford Univ. Press.
  • [45] van Dyk, D.A. (2014). The role of statistics in the discovery of a higgs boson. Annu. Rev. Statist. Appl. 1 41–59.
  • [46] Wang, H. (2004). Improved estimation of accuracy in simple hypothesis versus simple alternative testing. J. Multivariate Anal. 90 269–281.
  • [47] Wang, H. (2006). Modified $p$-value of two-sided test for normal distribution with restricted parameter space. Comm. Statist. Theory Methods 35 1361–1374.
  • [48] Wang, H. (2007). Modified $p$-values for one-sided testing in restricted parameter spaces. Statist. Probab. Lett. 77 625–631.
  • [49] Wang, Z. and Klir, G.J. (2009). Generalized Measure Theory. IFSR International Series on Systems Science and Engineering 25. New York: Springer.
  • [50] Wendell, J.P. and Schmee, J. (1996). Exact inference for proportions from a stratified finite population. J. Amer. Statist. Assoc. 91 825–830.
  • [51] Xie, M. and Singh, K. (2013). Confidence distribution, the frequentist distribution estimator of a parameter: A review. Int. Stat. Rev. 81 3–39.
  • [52] Zhang, T. and Woodroofe, M. (2003). Credible and confidence sets for restricted parameter spaces. J. Statist. Plann. Inference 115 479–490.
  • [53] Zhang, Z. and Zhang, B. (2013). A likelihood paradigm for clinical trials. J. Stat. Theory Pract. 7 157–177.