Chapter

# Defuzzification of a Fuzzy p-value by the Signed Distance: Application on Real Data

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## Abstract

We develop a fuzzy hypothesis testing approach where we consider the fuzziness of data and the fuzziness of the hypotheses as well. We give the corresponding fuzzy p-value with its $$\alpha$$-cuts. In addition, we use the so-called “signed distance” operator to defuzzify this p-value and we provide the convenient decision rule. Getting a defuzzified p-value and being able to interpret it can be of good use in many situations. We illustrate our testing procedure by a detailed numerical example where we study a right one-sided fuzzy test and compare it with a classical one. We close the paper by an application of the method on a survey from the financial place of Zurich, Switzerland. We display the decisions related to tests on the mean made on a set of variables of the sample. Both fuzzy and classical tests are conducted. One of our main findings is that despite the fact that each of both approaches have a different decision rule in terms of interpretation, the decisions made are by far the same. In this perspective, we can state that the fuzzy testing procedure can be seen as a generalization of the classical one.

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Statistical hypothesis testing is very important for finding decisions in practical problems. Usually, the underlying data are assumed to be precise numbers, but it is much more realistic in general to consider fuzzy values which are non-precise numbers. In this case the test statistic will also yield a non-precise number. This article presents an approach for statistical testing at the basis of fuzzy values by introducing the fuzzy p-value. It turns out that clear decisions can be made outside a certain interval which is determined by the characterizing function of the fuzzy p-values. Copyright Springer-Verlag 2004
A new approach of testing fuzzy hypotheses by confidence intervals and defuzzification of the fuzzy decision by the signed distance
• R Berkachy
• L Donzé
R. Berkachy and L. Donzé, "A new approach of testing fuzzy hypotheses by confidence intervals and defuzzification of the fuzzy decision by the signed distance," Under Review, 2018.