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Statistical Methods for Fuzzy Data by Reinhard Viertl

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Chapter 5

a.

b. The two inequalities (5.1) and (5.2) are consequences of the definition of the δ-cuts of the fuzzy relative frequencies. If the support of a fuzzy observation has non-empty intersection with two classes Kj and Kj + 1, then this observation is counted for n, δ(Kj) and for n, δ(Kj + 1). This observation is counted twice in the sum n(Kj) + n, δ(Kj + 1). If taking the union KjKj + 1 for n, δ(KjKj + 1) the observation is counted only once. From this inequality (5.1) is obtained. For the lower boundaries n, δ(Kj) and n, δ(Kj + 1), and n, δ(KjKj + 1) it is possible that the ...

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