Cramér-Rao Lower Bound for Fuzzy-Valued Random Variables
DOI:
https://doi.org/10.17713/ajs.v35i4.357Abstract
In some point estimation problems, we may confront imprecise (fuzzy) concepts. One important case is a situation where all observations are fuzzy rather than crisp. In this paper, using fuzzy set theory, we define a fuzzy-valued random variable, a fuzzy unbiased estimator, a fuzzy exponential family, and then we state and prove a Cramér-Rao lower bound for such fuzzy estimators. Finally, we give some examples.
References
Ash, R. B., and Doleans-Dade, C. A. (2000). Probability and Measure Theory (2nd ed.). Academic Press.
Billingsley, P. (1995). Probability and Measure (3rd ed.). John Wiley & Sons.
Buckley, J. J. (1985). Fuzzy decision making with data: Applications to statistics. Fuzzy Sets and Systems, 16, 139-147.
Casals, M. R., Gil, M. R., and Gil, P. (1986). On the use of Zadeh’s probabilistic definition for testing statistical hypotheses from fuzzy information. Fuzzy Sets and Systems, 20, 175-190.
Casella, G., and Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury Press.
Chung, K. L. (2000). A Course in Probability Theory (2nd ed.). Academic Press.
Coral, N., and Gil, M. A. (1984). The minimum inaccuracy fuzzy estimation: An extension of the maximum likelihood principle. Stochastica, 6, 63-81.
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Volume I (3rd ed.). John Wiley & Sons.
Freund, J. E. (1992). Mathematical Statistics (5th ed.). Prentice Hall.
Gertner, G. Z., and Zhu, H. (1996). Bayesian estimation in forest surveys when samples or prior information are fuzzy. Fuzzy Sets and Systems, 77, 277-290.
Gil, M. A., Corral, N., and Gil, P. (1985). The fuzzy decision problem: An approach to the point estimation problem with fuzzy information. European Journal of Operation
Research, 22, 26-34.
Hogg, R. V., and Craig, A. T. (1995). Introduction to Mathematical Statistics (5th ed.). Eaglewood Cliffs, NJ: Prentice-Hall.
Juninig, O., and Wang, G. (1989). Fuzzy random variables and their probability characters. Journal of Harbin Architectural and Civil Engineering Institute, 12, 1-10.
Kruse, R. (1984). Statistical estimation with linguistic data. Information Sciences, 33, 197-207.
Kruse, R., and Meyer, K. D. (1987). Statistics with Vague Data. Dordrech, Netherlands: Reidel Pub. Comp.
Kwakernaak, H. (1978). Fuzzy random variables: Definition and theorems. Information Sciences, 15, 1-29.
Lehmann, E. L., and Casella, G. (1998). Theory of Point Estimation. Springer-Verlag.
Liu, B. (2004). Uncertainty Theory: An Introduction to Its Axiomatic Foundation. Heidelberg: Physica-Verlag.
López-Díaz, M., and Gil, M. A. (1997). Constructive definitions of fuzzy random variables. Statistics and Probability Letters, 36, 135-144.
M. López-Díaz, M. A. G. (1998). Approximately integrable bounded fuzzy random variable in terms of the “generalized” Hausdorff metric. Information Sciences, 104, 279-291.
Mood, A. M., Graybill, F. A., and Boes, D. C. (1974). Introduction to the Theory of Statistics (3rd ed.). McGraw-Hill.
Okuda, T. (1987). A statistical treatment of fuzzy observations: Estimation problems. 51-55. (Preprints of the Second IFSA Congress)
Puri, M. L., and Ralescu, D. A. (1986). Fuzzy random variables. Journal of Mathematical Analysis and Applications, 114, 409-422.
Ralescu, D. A. (1995). Fuzzy random variable revisited. In Proc. IFES’ 95, Vol. 2 (p. 993-1000).
Ross, S. M. (2002). A First Course in Probability (6th ed.). Prentice-Hall.
Shao, J. (1998). Mathematical Statistics. New York: Springer-Verlag.
Yao, J. S., and Hwang, C. M. (1996). Point estimation for the n sizes of random sample with one vague data. Fuzzy Sets and Systems, 80, 205-215.
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338-353.
Zadeh, L. A. (1968). Probability measure of fuzzy events. Journal of Mathematical Analysis and Applications, 23, 421-427.
Downloads
Published
Issue
Section
License
The Austrian Journal of Statistics publish open access articles under the terms of the Creative Commons Attribution (CC BY) License.
The Creative Commons Attribution License (CC-BY) allows users to copy, distribute and transmit an article, adapt the article and make commercial use of the article. The CC BY license permits commercial and non-commercial re-use of an open access article, as long as the author is properly attributed.
Copyright on any research article published by the Austrian Journal of Statistics is retained by the author(s). Authors grant the Austrian Journal of Statistics a license to publish the article and identify itself as the original publisher. Authors also grant any third party the right to use the article freely as long as its original authors, citation details and publisher are identified.
Manuscripts should be unpublished and not be under consideration for publication elsewhere. By submitting an article, the author(s) certify that the article is their original work, that they have the right to submit the article for publication, and that they can grant the above license.
How to Cite
