4.3 Cumulative sum for fuzzy data

In the case of classical samples x1,…,xn with xi ∈ of one-dimensional quantities, cumulative sums are defined by

and normalized cumulative sums Sn(·) are defined by

Such cumulative sums give the proportion of observations which are not larger than a given real value x.

Remark 4.2:

For (idealized) precise observations xi ∈ the cumulative sum coincides with the empirical distribution function , defined by its values

For fuzzy observations given as fuzzy numbers xi*, i = 1(1)n, the definition of cumulative sums has to be adapted. This adapted cumulative sum Snad(·) is defined using the characterizing functions ξi(·) of xi*, i = 1(1)n, which are assumed to be integrable, by

Remark 4.2:

Snad(x) is the proportion of fuzzy observations less than or equal to ...

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