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... The signed distance measure was firstly used in the context of ranking fuzzy numbers by Yao and Wu [23]. It has also served in some other contexts: Berkachy and Donzé [3] used it in the assessment of linguistic questionnaires; Berkachy and Donzé [6] used it in hypotheses testing; etc. Although this measure is considered to be simple in terms of computations, it has interested specialists because of its directionality. ...

... First of all, we would like to construct the fuzzy confidence intervals by the likelihood ratio method at the confidence level 95% for the variables PW5020 and HQ5010 as previously discussed. Each modality of these variables is modelled by a triangular symmetrical fuzzy number of spread 2. In other words, the value 3 is for example modelled by the triangular fuzzy number (2,3,4). It is the same for all the other modalities of both variables. ...

We propose a practical procedure of construction of fuzzy confidence intervals by the likelihood method where the observations and the hypotheses are considered to be fuzzy. We use the bootstrap technique to estimate the distribution of the likelihood ratio. The chosen bootstrap algorithm consists on randomly drawing observations by preserving the location and dispersion measures of the original fuzzy data set. A metric $$d_{SGD}^{\theta ^{\star }}$$ d SGD θ ⋆ based on the well-known signed distance measure is considered in this case. We expose a simulation study to investigate the influence of the fuzziness of the computed maximum likelihood estimator on the constructed confidence intervals. Based on these intervals, we introduce a hypothesis test for the equality of means of two groups with its corresponding decision rule. The highlight of this paper is the application of the defended approach on the Swiss SILC Surveys. We empirically investigate the influence of the fuzziness vs. the randomness of the data as well as of the maximum likelihood estimator on the confidence intervals. In addition, we perform an empirical analysis where we compare the mean of the group “Swiss nationality” to the group “Other nationalities” for the variables Satisfaction of health situation and Satisfaction of financial situation.

... Afterwards, based on the work of Grzegorzewki [8], Berkachy & Donzé [9] proposed the signed distance as a defuzzification operator, and gave its use on fuzzy decisions. We remind that this method has been extensively used in other contexts such as evaluating linguistic questionnaires, as seen in Berkachy & Donzé [10]. ...

... Yao & Wu [12] and Lin & Lee [13] mainly described the signed distance defuzzification method. Afterwards, Berkachy & Donzé [10] have used it in another context, the evaluation of linguistic questionnaires. Regarding its nice properties, this method will be implemented in our test procedure for defuzzifying the fuzzy p-values. ...

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.

... In this work, the purpose is to defuzzify a fuzzy decision by the signed distance method defined in Definition 3.3. As instance, (Berkachy and Donzé, 2016a) used this defuzzification method in order to aggregate linguistic questionnaires. Indeed, inspired by these aggregations, we consider the fuzzy test statistic as a linguistic variable decomposed into 2 linguistic terms, i.e. rejection or acceptance of a given hypothesis, with distances d(L 1 ,0) and d(L 2 ,0) related to the corresponding fuzzy numbers {L q } 2 q=1 . ...

... From (Berkachy and Donzé, 2016a) and the previous section, we define the signed distance defuzzification operator O SGD of the decision D in the following manner: ...

Linguistic questionnaires have gained lots of attention in the last decades. They are a prominent tool used in many fields to convey the opinion of people on different subjects. In this chapter, we propose a model for the assessment of linguistic questionnaires describing the global and individual evaluations, where we suppose that the sample weights and the missingness are both allowed. For the problem of missingness, we show a method based on the readjustment of weights. We should clarify that the proposed approach is not a correction for the missingness in the sample as widely known in survey statistics. We give the expressions of the individual and global evaluations, followed by the description of the indicators of information rate related to missing answers. The model is after illustrated by a numerical application related to the Finanzplatz data set. The objective of this empirical application is to clearly see that the obtained individual evaluations can be treated similarly to any data set in the classical theory. In addition, we will empirically remark that the obtained distributions tend to be normally distributed. Afterward, we perform different analyses by simulations on the statistical measures of these distributions. We compare the individual evaluations with respect to a variety of distances, in order to see the influence of the symmetry of the modelling fuzzy numbers and the sample sizes on different statistical measures. We close the chapter by a comparison between the evaluations by the defended model, and the ones obtained through a usual fuzzy system using different defuzzification operators. Interesting findings of this chapter are that corresponding statistical measures are independent from the sample sizes, and that the use of the traditional fuzzy rule-based systems is not always the most convenient tool when non-symmetrical modelling shapes are used, contrariwise to the defended approach. Our approach by the individual evaluations is in such situations suggested.

Testing hypotheses could sometimes benefit from the fuzzy context of data or from the lack of precision in specifying the hypotheses. A fuzzy approach is therefore needed for reflecting the right decision regarding these hypotheses. Different methods of testing hypotheses in a fuzzy environment have already been presented. On the basis of the classical approach, we intend to show how to accomplish a fuzzy test. In particular, we consider that the fuzziness does not only come from data but from the hypotheses as well. We complete our review by explaining how to defuzzify the fuzzy test decision by the signed distance method in order to obtain a crisp decision. The detailed procedures are presented with numerical examples of real data. We thus present the pros and cons of both the fuzzy and classical approaches. We believe that both approaches can be used in specific conditions and contexts, and guidelines for their uses should be identified.

With today’s information overload, it has become increasingly difficult to analyze the huge amounts of data and to generate appropriate management decisions. Furthermore, the data are often imprecise and will include both quantitative and qualita- tive elements. For these reasons, it is important to extend traditional decision making processes by adding intuitive reasoning, human subjectivity and imprecision.
In the age of Big Data, decision making processes for economy and society have to deal with uncertainty, vagueness, and imprecision. Besides Volume, Variety, and Velocity, two others V’s for Veracity and Value have also to be taken into consideration. Therefore, the application of fuzzy sets and fuzzy logic becomes a hot topic.
In 2008, the Department for Informatics at the University of Fribourg, Switzer- land founded its Research Center for Fuzzy Management Methods (FMM = FM2), often only called FMsquare. Later on, the International Research Book Series for FMsquare was launched by Springer, where researchers published in fuzzy-based reputation management, fuzzy classification of online customers, inductive fuzzy classification for marketing analytics, fuzzy data warehousing for performance measurement, using intuitionistic fuzzy sets for service level engineering, building a knowledge carrier based on granular computing, or a fuzzy-based recommender system for political communities, among others.

The central moments of a random variable are extensively used to understand the characteristics of distributions in classical statistics. It is well known that the second central moment of a given random variable is simply its variance. When fuzziness in data occurs, the situation becomes much more complicated. The central moments of a fuzzy random variable are often very difficult to be calculated because of the analytical complexity associated with the product of two fuzzy numbers. An estimation is needed. Our research showed that the so-called signed distance is a great tool for this task. The main contribution of this paper is to present the central moments of a fuzzy random variable using this distance. Furthermore, since we are interested in the statistical measures of the distribution, particularly the variance, we put an attention on its estimation using the signed distance. Using this distance in approximating the square of a fuzzy difference, we can get an unbiased estimator of the variance. Finally, we prove that in some conditions our methodology related to the signed distance returns an exact crisp variance.

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