Vague data

Joint Treatment of Imprecision and Randomness in Uncertainty Propagation

“…To evaluate the average imprecision about T , we can estimate an average fuzzy number π T mean with [20]:…”

“…Random sampling is resumed and the process is performed in an iterative fashion to obtain a sample (π T 1 , ..., π T m ) of fuzzy subsets (see Fig.1). T (X) becomes a fuzzy random variable [20]. In the case where dependencies are 0 1 frag replacements Model result: Random Fuzzy Numbers Fig.…”

Possibility/Necessity-Based Probabilistic Expectation Models for Linear Programming Problems with Discrete Fuzzy Random Variables

“…A fuzzy random variable was firstly defined by Kwakernaak [24] as an extended concept of random variables in the sense that the realized values for given events or scenarios are not real but fuzzy numbers. Kruse and Meyer [35] provided some concepts similar to the model by Kwakernaak. Puri and Ralescu [25] defined fuzzy random variables as random fuzzy sets and developed a mathematical basis of fuzzy random variables with Klemment [23].…”

Cramér-Rao Lower Bound for Fuzzy-Valued Random Variables

“…Some methods in descriptive statistics with vague data and some aspects of statistical inference is proposed in Kruse and Meyer (1987). 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.…”

“…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). For more details about ordinary point estimation problems (with crisp data), see Casella and Berger (2002), Freund (1992), Hogg and Craig (1995), Lehmann and Casella (1998), Mood, Graybill, and Boes (1974), and Shao (1998).…”