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6.3 Smoothed empirical distribution function

Another possibility to adapt the concept of the empirical distribution function to the situation of fuzzy data is the so-called smoothed empirical distribution function nsm(·).

The values nsm(x) of the smoothed empirical distribution function based on fuzzy data x1*,…, xn* with characterizing functions ξ1(·),…,ξn(·) are defined by generalizing the classical empirical distribution function n(·) from Section 4.3 (cf. Remark 4.2).

provided that all characterizing functions ξ1(·),…,ξn(·) are integrable with finite integral

In case the sample contains some precise values, say x1,…, xl and the rest are fuzzy values xl+1*,…, xn* with characterizing functions ξl+1(·),…,ξn(·), the smoothed empirical distribution function is defined by

In Figure 6.3 a so-called mixed sample and the corresponding smoothed empirical distribution function are depicted.

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