Fuzzy random variables—I. definitions and theorems

Kwakernaak, Huibert

doi:10.1016/0020-0255(78)90019-1

Cited by 1,118 publications

(333 citation statements)

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“…This type of construct was proposed in the setting of probability spaces under the name fuzzy random variables (Kwakernaak (1978(Kwakernaak ( , 1979Puri and Ralescu, 1986). See Gebhardt et al (1998)'s andRalescu (1999)'s Chapters in the Handbook of Fuzzy Sets for extensive surveys on this notion, respectively from a statistical and mathematical point of view.…”

Section: Probabilistic Extensions Of Fuzzy Setsmentioning

confidence: 99%

Fuzzy Sets: History and Basic Notions

Dubois¹,

Ostasiewicz

Prade³

2000

The Handbooks of Fuzzy Sets Series

139

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Abstract. This paper is an introduction to fuzzy set theory. It has several purposes. First, it tries to explain the emergence of fuzzy sets from an historical perspective. Looking back to the history of sciences, it seems that fuzzy sets were bound to appear at some point in the 20th century. Indeed, Zadeh's works have cristalized and popularized a concern that has appeared in the first half of the century, mainly in philosophical circles. Another purpose of the paper is to scan the basic definitions in the field, that are required for a proper reading of the rest of the volume, as well as the other volumes of the Handbooks of Fuzzy Sets Series. This Chapter also contains a discussion on operational semantics of the generally too abstract notion of membership function. Lastly, a survey of variants of fuzzy sets and related matters is provided.

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Section: Probabilistic Extensions Of Fuzzy Setsmentioning

confidence: 99%

Fuzzy Sets: History and Basic Notions

Dubois¹,

Ostasiewicz

Prade³

2000

The Handbooks of Fuzzy Sets Series

139

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“…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), López-Díaz and Gil (1997), M. López-Díaz (1998), and Liu (2004. 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).…”

Section: Introductionmentioning

confidence: 99%

Cramér-Rao Lower Bound for Fuzzy-Valued Random Variables

Torabi

2016

AJS

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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.

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“…In the literature, it contains different definitions of fuzzy random variables. Kwakernaak has proposed the concept of fuzzy random variable for the first time [15]. Further, it was used by Kruse and Meyer [16].…”

Section: Introductionmentioning

confidence: 99%