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6.2 Fuzzy empirical fractiles

For classical empirical distribution functions n(·) the empirical fractiles are defined in the following way: Let p∈(0; 1) then the p-fractile xp is given by In the case of fuzzy empirical distribution functions n*(·) generalized empirical fractiles are fuzzy numbers. The characterizing function or the δ-cuts of the fuzzy empirical fractiles are defined in the following way:

For p ∈ (0, 1) the lower and upper δ-level curves δ,L (·) and δ,U (·) are used to define the δ-cuts of the corresponding fuzzy fractile qp*. The δ-cut Cδ (qp*) is defined by The method is explained in Figure 6.2.

Figure 6.2 Construction of Cδ(qp*). Remark 6.1:

The characterizing function ξq*p(·) of the fuzzy empirical quantile qp* is obtained by Lemma 2.1.

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