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