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Statistical Methods for Fuzzy Data by Reinhard Viertl

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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).

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

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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

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

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