Statistical Methods for Fuzzy Data

6

Empirical distribution functions

The classical empirical distribution function for n precise data points x1,…, xn is defined by

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

for all δ ...