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