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Abstract
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.
Highlights
Point estimation in the traditional statistical inference is based on crispness of data, random variables, and so on
Some methods in descriptive statistics with vague data and some aspects of statistical inference is proposed in Kruse and Meyer (1987)
We offer two proofs: Proof (1): By Theorem 5.2, this family belongs to the fuzzy exponential family, because f(x; θ) = (0.1 + 0.8θ) exp
Summary
Point estimation in the traditional statistical inference is based on crispness of data, random variables, and so on. Fuzzy random variables were introduced by Kwakernaak (1978), Puri and Ralescu (1986) as a generalization of compact random sets, Kruse and Meyer (1987) and were developed by others such as Juninig and Wang (1989), Ralescu (1995), Lopez-Dıaz and Gil (1997), M. Some aspects of point estimation problems with fuzzy data are discussed in Yao and Hwang (1996), Buckley (1985), Coral and Gil (1984), Gertner and Zhu (1996), Gil, Corral, and Gil (1985), Kruse (1984), Kruse and Meyer (1987), and Okuda (1987).
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