Ordered fuzzy random variables: Definition and the concept of normality

The concept of fuzzy random variable combines two sources of uncertainty: randomness and fuzziness, whereas the model of ordered fuzzy numbers provides a representation of inaccurate quantitative data, and is an alternative to the standard fuzzy numbers model proposed by Zadeh. This paper develops the model of ordered fuzzy numbers by defining the concept of fuzzy random variables for these numbers, called further ordered fuzzy random variables. Thanks to the well-defined arithmetic of ordered fuzzy numbers (existence of neutral and opposite elements) and the introduced ordered fuzzy random variables; it becomes possible to construct fully fuzzy stochastic time series models such as e.g., the autoregressive model or the GARCH model in the form of classical equations, which can be estimated using the least-squares or the maximum likelihood method. Furthermore, the concept of normality of ordered fuzzy random variables and the method to generate pseudo-random ordered fuzzy variables with normal distribution are introduced. (C) 2020 Elsevier Inc. All rights reserved.