13.2 Fuzzy p-values

Let η(·) be the characterizing function of the fuzzy value t* = t(x1*, … , xn*) of a test statistic T = t(x1, … , xn) based on the fuzzy sample x1*, … , xn*.

In applications the support supp [η(·)] = {x ∈ : η(x) > 0} is usually a bounded subset of .

Considering the δ-cuts of t* for δ ∈ (0; 1] and denoting them by

the corresponding fuzzy p-value p* is defined via its δ-cuts Cδ(p*) for δ ∈ (0; 1] in the following way:

For a one-sided test with decision rule for T ≤ tcr reject hypothesis, the δ-cuts of the corresponding fuzzy p-value p* are defined by

with t1(δ) and t2(δ) from above.

For one-sided tests with decision rule for T ≥ tcr reject the hypothesis, the δ-cults of the corresponding fuzzy p-value p* are defined by

In the case of two-sided tests first it has to be decided on which side of the median m of the distribution of T the main part of the amount of fuzziness of t* is located. Therefore one has to compute the area under the characterizing function η(·) of ...

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