×

zbMATH — the first resource for mathematics

Discussion: Foundations of statistical inference, revisited. (English) Zbl 1332.62024
Summary: This is an invited contribution to the discussion on Professor D. Mayo’s paper [ibid. 29, No. 2, 227–239 (2014; Zbl 1332.62025)]. Mayo clearly demonstrates that statistical methods violating the likelihood principle need not violate either the sufficiency or conditionality principle, thus refuting A. Birnbaum’s claim [J. Am. Stat. Assoc. 57, 269–306, 307–326 (1962; Zbl 0107.36505)]. With the constraints of Birnbaum’s theorem lifted, we revisit the foundations of statistical inference, focusing on some new foundational principles, the inferential model framework, and connections with sufficiency and conditioning.

MSC:
62A01 Foundations and philosophical topics in statistics
62D05 Sampling theory, sample surveys
62B05 Sufficient statistics and fields
PDF BibTeX XML Cite
Full Text: DOI Euclid arXiv
References:
[1] Birnbaum, A. (1962). On the foundations of statistical inference. J. Amer. Statist. Assoc. 57 269-326. · Zbl 0107.36505
[2] Chiang, A. K. L. (2001). A simple general method for constructing confidence intervals for functions of variance components. Technometrics 43 356-367.
[3] Dempster, A. P. (2008). The Dempster-Shafer calculus for statisticians. Internat. J. Approx. Reason. 48 365-377. · Zbl 1274.62053
[4] Evans, M. (2013). What does the proof of Birnbaum’s theorem prove? Electron. J. Stat. 7 2645-2655. · Zbl 1294.62002
[5] Evans, M. J., Fraser, D. A. S. and Monette, G. (1986). On principles and arguments to likelihood. Canad. J. Statist. 14 181-199. · Zbl 0607.62002
[6] Fisher, R. A. (1930). Inverse probability. Math. Proc. Cambridge Philos. Soc. 26 528-535. · JFM 56.1083.05
[7] Fisher, R. A. (1973). Statistical Methods and Scientific Inference , 3rd ed. Hafner Press, New York. · Zbl 0281.62002
[8] Fraser, D. A. S. (1968). The Structure of Inference . Wiley, New York. · Zbl 0164.48703
[9] Fraser, D. A. S. (2004). Ancillaries and conditional inference. Statist. Sci. 19 333-369. · Zbl 1100.62534
[10] Fraser, D. A. S. (2011). Is Bayes posterior just quick and dirty confidence? Statist. Sci. 26 299-316. · Zbl 1246.62040
[11] Ghosh, M., Reid, N. and Fraser, D. A. S. (2010). Ancillary statistics: A review. Statist. Sinica 20 1309-1332. · Zbl 1200.62001
[12] Hannig, J. (2009). On generalized fiducial inference. Statist. Sinica 19 491-544. · Zbl 1168.62004
[13] Hannig, J. (2013). Generalized fiducial inference via discretization. Statist. Sinica 23 489-514. · Zbl 1379.62002
[14] Lindley, D. V. (1958). Fiducial distributions and Bayes’ theorem. J. R. Stat. Soc. Ser. B Stat. Methodol. 20 102-107. · Zbl 0085.35503
[15] Liu, C. and Martin, R. (2015). Inferential Models : Reasoning with Uncertainty. Monographs in Statistics and Applied Probability Series . Chapman & Hall, London. · Zbl 1377.62018
[16] Martin, R. and Liu, C. (2013). Inferential models: A framework for prior-free posterior probabilistic inference. J. Amer. Statist. Assoc. 108 301-313. · Zbl 06158344
[17] Martin, R. and Liu, C. (2014a). Conditional inferential models: Combining information for prior-free probabilistic inference. J. R. Stat. Soc. Ser. B. Stat. Methodol. To appear. Full-length preprint version at arXiv:
[18] Martin, R. and Liu, C. (2014b). A note on \(p\)-values interpreted as plausibilities. Statist. Sinica . To appear. Available at arXiv: · Zbl 06497320
[19] Martin, R., Zhang, J. and Liu, C. (2010). Dempster-Shafer theory and statistical inference with weak beliefs. Statist. Sci. 25 72-87. · Zbl 1328.62040
[20] Shafer, G. (1976). A Mathematical Theory of Evidence . Princeton Univ. Press, Princeton, NJ. · Zbl 0359.62002
[21] Taraldsen, G. and Lindqvist, B. H. (2013). Fiducial theory and optimal inference. Ann. Statist. 41 323-341. · Zbl 1347.62019
[22] Weerahandi, S. (1993). Generalized confidence intervals. J. Amer. Statist. Assoc. 88 899-905. · Zbl 0785.62029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.