Abstract
In this paper, the transitivity properties of reciprocal relations, also called probabilistic relations, are investigated within the framework of cycle-transitivity. Interesting types of transitivity are highlighted and shown to be realizable in applications. For example, given a collection of random variables (X k )k ∈ I, pairwisely coupled by means of a same copula C ∈ {T M , T P , T L }, the transitivity of the reciprocal relation Q defined by \(Q (X_i,X_j) = {\rm Prob}\{X_i X_j\} + 1/2 {\rm\ Prob}\{X_i=X_j\}\) can be characterized within the cycle- transitivity framework. Similarly, given a poset (P, ≤ ) with P = {x 1, ..., x n }, the transitivity of the mutual rank probability relation Q P , where Q P (X i ,X j ) denotes the probability that x i precedes x j in a random linear extension of P, is characterized as a type of cycle-transitivity for which no realization had been found so far.
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De Baets, B., De Meyer, H., De Loof, K. (2008). The Omnipresence of Cycle-Transitivity in the Comparison of Random Variables. In: Dubois, D., Lubiano, M.A., Prade, H., Gil, M.Á., Grzegorzewski, P., Hryniewicz, O. (eds) Soft Methods for Handling Variability and Imprecision. Advances in Soft Computing, vol 48. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85027-4_36
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DOI: https://doi.org/10.1007/978-3-540-85027-4_36
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