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Statistical Methods for Fuzzy Data by Reinhard Viertl

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6.1 Fuzzy valued empirical distribution function

For fuzzy samples x1*,…, xn* with corresponding characterizing functions ξ1(·),…,ξn(·) the counterpart to the empirical distribution function is a fuzzy valued function n*(·) defined on whose lower and upper δ-level functions δ,L(·) and δ,U(·) are defined via the δ-cuts Cδ(xi*) of the fuzzy observations xi*, which are assumed to be fuzzy intervals.

For fixed x the values δ,L(x) and δ,U(x) are defined by

Unnumbered Display Equation

for all δ ∈ [0; 1].

Lemma 6.1:

For fuzzy intervals x1*,…, xn* whose δ-cuts are given by

Unnumbered Display Equation

the corresponding δ-level functions of the fuzzy valued ...

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