Random intervals as a model for imprecise information. (English) Zbl 1079.60012
For an initial probability space \((\Omega, A, P)\) and a random variable \(U: \Omega \rightarrow R\) modelling the behaviour of some elements \(\omega \in\Omega\), the imprecision or the missing data in the observation of the \(U(\omega)\) values entail that, for any \(\omega \in \Omega\), \(U(\omega)\) belongs to the random (interval) set \(\Gamma(\omega) = [A(\omega), B(\omega)\)]. \(U\) belongs to the class of random variables whose values are included in the random interval \(\Gamma\), this class being denoted by \(P(\Gamma)\), and interpreted as the class of probability distributions of the measurable selections of the random (interval) set \(\Gamma\). In the same time, the probability induced by \(U\) is bounded between the upper and lower probabilities of the random interval \(\Gamma\), and the class of all these probabilities is denoted by \(M(P^*)\).
The aim of this paper is to investigate the relationship between \(P(\Gamma)\) and \(M(P^*)\), as two models for uncertainty and imprecision of the information about the probability distribution of the random variable \(U\), when \(\Gamma\) is a random interval. While \(P(\Gamma)\) is the most precise of the two models, since \(P(\Gamma) \subseteq M(P^*)\), the class \(M(P^*)\) is more interesting from an operational point of view, being convex, closed for some situations, and uniquely determined by the values of \(P^*\). The following main results are proved within the paper: (a) Given a random (closed or open) interval \(\Gamma\) defined on a non-atomic probability space, the closures of the classes \(P(\Gamma)\) and \(M(P^*)\), under the topology of weak convergence, coincide. (b) \(P(\Gamma)\) is a proper subset of \(M(P^*)\) when either of the distribution functions of the extremes of the random interval is continuous. Examples showing that \(P(\Gamma)\) and \(M(P^*)\) are not equivalent in general are provided. (c) The relationship between random intervals and fuzzy numbers is examined and discussed.
The aim of this paper is to investigate the relationship between \(P(\Gamma)\) and \(M(P^*)\), as two models for uncertainty and imprecision of the information about the probability distribution of the random variable \(U\), when \(\Gamma\) is a random interval. While \(P(\Gamma)\) is the most precise of the two models, since \(P(\Gamma) \subseteq M(P^*)\), the class \(M(P^*)\) is more interesting from an operational point of view, being convex, closed for some situations, and uniquely determined by the values of \(P^*\). The following main results are proved within the paper: (a) Given a random (closed or open) interval \(\Gamma\) defined on a non-atomic probability space, the closures of the classes \(P(\Gamma)\) and \(M(P^*)\), under the topology of weak convergence, coincide. (b) \(P(\Gamma)\) is a proper subset of \(M(P^*)\) when either of the distribution functions of the extremes of the random interval is continuous. Examples showing that \(P(\Gamma)\) and \(M(P^*)\) are not equivalent in general are provided. (c) The relationship between random intervals and fuzzy numbers is examined and discussed.
Reviewer: Neculai Curteanu (Iaşi)
MSC:
| 60B10 | Convergence of probability measures |
| 60F05 | Central limit and other weak theorems |
| 60A10 | Probabilistic measure theory |
| 28E10 | Fuzzy measure theory |
| 94D05 | Fuzzy sets and logic (in connection with information, communication, or circuits theory) |
| 65G40 | General methods in interval analysis |
Keywords:
random sets; measurable selections; Dempster-Shafer upper and lower probabilities; models for uncertainty and imprecision; non-atomic probability space; closures of probability spaces; fuzzy numbers; weak convergence topologyReferences:
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