Statistical inference based on the possibility and belief measures
HTML articles powered by AMS MathViewer
- by Yuan Yan Chen PDF
- Trans. Amer. Math. Soc. 347 (1995), 1855-1863 Request permission
Abstract:
In statistical inference, we infer the population parameter based on the realization of sample statistics. This can be considered in the framework of inductive inference. We showed, in Chen (1993), that if we measure a parameter by the possibility (or belief) measure, we can have an inductive inference similar to the Bayesian inference in belief update. In this article we apply this inference to statistical estimation and hypotheses evaluation (testing) for some parametric models, and compare them to the classical statistical inferences for both one-sample and two-sample problems.References
- G. A. Barnard, R. A. Fisher—a true Bayesian?, Internat. Statist. Rev. 55 (1987), no. 2, 183–189 (English, with French summary). MR 963339, DOI 10.2307/1403194
- Allan Birnbaum, On the foundations of statistical inference, J. Amer. Statist. Assoc. 57 (1962), 269–326. MR 138176, DOI 10.1080/01621459.1962.10480660
- Yuan Yan Chen, Bernoulli trials: from a fuzzy measure point of view, J. Math. Anal. Appl. 175 (1993), no. 2, 392–404. MR 1219183, DOI 10.1006/jmaa.1993.1178 J. Cohen (1970), The implications of induction, Methuen, London.
- A. W. F. Edwards, Likelihood, Expanded edition, Johns Hopkins University Press, Baltimore, MD, 1992. Revised reprint of the 1972 original. MR 1191161
- Thomas S. Ferguson, A Bayesian analysis of some nonparametric problems, Ann. Statist. 1 (1973), 209–230. MR 350949 A. Fisher (1935), The logic of inductive inference, J. Roy. Statist. Soc. 98, 39-54. —(1936), Uncertain inference, Proc. Amer. Acad. Arts and Sciences 71, 245-258. —(1956), Statistical methods and scientific inference, Oliver & Boyd, Edinburgh.
- I. J. Good, Probability and the Weighing of Evidence, Charles Griffin & Co., Ltd., London; Hafner Publishing Co., New York, N. Y., 1950. MR 0041366
- Ian Hacking, The emergence of probability, Cambridge University Press, London-New York, 1975. A philosophical study of early ideas about probability, induction and statistical inference. MR 0467864
- H. Ichihashi, H. Tanaka, and K. Asai, Fuzzy integrals based on pseudo-additions and multiplications, J. Math. Anal. Appl. 130 (1988), no. 2, 354–364. MR 929941, DOI 10.1016/0022-247X(88)90311-3
- K. Popper and D. W. Miller, Why probabilistic support is not inductive, Philos. Trans. Roy. Soc. London Ser. A 321 (1987), no. 1562, 569–591. MR 892293, DOI 10.1098/rsta.1987.0033 L. S. Shackle (1961), Decision, order and time in human affairs, Cambridge Univ. Press, Cambridge.
- Glenn Shafer, A mathematical theory of evidence, Princeton University Press, Princeton, N.J., 1976. MR 0464340, DOI 10.1515/9780691214696
- Glenn Shafer, Belief functions and parametric models, J. Roy. Statist. Soc. Ser. B 44 (1982), no. 3, 322–352. With discussion. MR 693232, DOI 10.1111/j.2517-6161.1982.tb01211.x
- Zi Xiao Wang, Fuzzy measures and measures of fuzziness, J. Math. Anal. Appl. 104 (1984), no. 2, 589–601. MR 766153, DOI 10.1016/0022-247X(84)90022-2
- L. A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978), no. 1, 3–28. MR 480045, DOI 10.1016/0165-0114(78)90029-5
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 347 (1995), 1855-1863
- MSC: Primary 62A10
- DOI: https://doi.org/10.1090/S0002-9947-1995-1285980-X
- MathSciNet review: 1285980