Abstract
In previous papers, the consequences of the “presence of fuzziness” in the experimental information on which statistical inferences are based were discussed. Thus, the intuitive assertion «fuzziness entails a loss of information» was formalized, by comparing the information in the “exact case” with that in the “fuzzy case”. This comparison was carried out through different criteria to compare experiments (in particular, that based on the “pattern” one, Blackwell's sufficiency criterion). Our purpose now is slightly different, in the sense that we try to compare two “fuzzy cases”. More precisely, the question we are interested in is the following: how will different “degrees of fuzziness” in the experimental information affect the sufficiency? In this paper, a study of this question is carried out by constructing an alternative criterion (equivalent to sufficiency under comparability conditions), but whose interpretation is more intuitive in the fuzzy case. The study is first developed for Bernoulli experiments, and the coherence with the axiomatic requirements for measures of fuzziness is also analyzed in such a situation. Then it is generalized to other random experiments and a simple example is examined.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bezdek, J. C. (1987). Partition structures: a tutorial, Analysis of Fuzzy Information, III, 81–107, CRC Press, Boca Raton, Florida.
Blackwell, D. A. (1951). Comparison of experiments, Proc. 2nd Berkeley Symp. on Math. Statist. Prob., 93–102, University of California Press, Berkeley.
Blackwell, D. A. (1953). Equivalent comparisons of experiments, Ann. Math. Statist., 24, 265–272.
DeLuca, A. and Termini, S. (1972). A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory, Inform. and Control, 20, 301–312.
Ferentinos, K. and Papaioannou, T. (1979). Loss of information due to groupings, Transactions of the Eighth Prague Conference on Information Theory, Statistical Decision Functions, Random Process, Vol. C, 87–94, Reidel, Dordrecht.
Gil, M. A. (1987). Fuzziness and loss of information in statistical problems, IEEE Trans. Systems Man Cybernet., 17, 1016–1025.
Gil, M. A. (1988a). On the loss of information due to fuzziness in experimental observations, Ann. Inst. Statist. Math., 40, 627–639.
Gil, M. A. (1988b). Probabilistic-possibilistic approach to some statistical problems with fuzzy experimental observations, Combining fuzzy imprecision with probabilistic uncertainty in decision making, Lecture Notes in Econom. and Math. Systems (eds. J.Kacprzyk and M.Fedrizzi), 310, 286–306, Springer, Berlin.
Gil, M. A., Corral, N. and Gil, P. (1988). The minimum inaccuracy estimates in χ2 tests for goodness of fit with fuzzy observations, J. Statist. Plann. Inference, 19, 95–115.
Kale, K. (1964). A note on the loss of information due to grouping of observations, Biometrika, 51, 495–497.
Klir, G. J. and Folger, T. A. (1988). Fuzzy Sets, Uncertainty and Information, Prentice Hall, New Jersey.
Kullback, S. (1968). Information Theory and Statistics, Dover, New York.
LeCam, L. (1964). Sufficiency and approximate sufficiency, Ann. Math. Statist., 35, 1419–1455.
LeCam, L. (1986). Asymptotic Methods in Statistical Decision Theory, Springer, Berlin.
Lehmann, E. L. (1988). Comparing location experiments, Ann. Statist., 16, 521–533.
Lindley, D. V. (1956). On a measure of the information provided by an experiment, Ann. Math. Statist., 27, 986–1005.
Okuda, T., Tanaka, H. and Asai, K. (1978). A formulation of fuzzy decision problems with fuzzy information, using probability measures of fuzzy events, Inform. and Control, 38, 135–147.
Papoulis, A. (1962). The Fourier Integral and Its Applications, McGraw-Hill, New York.
Raiffa, H. and Schlaifer, R. (1961). Applied Ststistical Decision Theory, The M.I.T. Press, Cambridge, Massachusetts.
Stone, M. (1961). Non-equivalent comparisons of experiments and their use for experiments involving location parameters, Ann. Math. Statist., 32, 326–332.
Tanaka, H., Okuda, T. and Asai, K. (1979). Fuzzy information and decision in statistical model, Advances in Fuzzy Sets Theory and Applications, 303–320, North-Holland, Amsterdam.
Zadeh, L. A. (1968). Probability measures of fuzzy events, J. Math. Anal. Appl., 23, 421–427.
Zadeh, L. A. (1978). Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1, 3–28.
Author information
Authors and Affiliations
Additional information
This research is supported in part by DGICYT Grant PS89-0169, NASA Grant NCC 2-275, and NSF PYI Grant DMC-84511622. Their financial support is gratefully acknowledged.
The author wrote this paper when she was a Research Associate of the Department of Electrical Engineering and Computer Science (Computer Science Division), University of California, Berkeley, CA 94720.
About this article
Cite this article
Gil, M.A. Sufficiency and fuzziness in random experiments. Ann Inst Stat Math 44, 451–462 (1992). https://doi.org/10.1007/BF00050698
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00050698