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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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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.

Fuzzy statistical methods appear to be well suited to situations where the data we are collecting are exposed to fuzziness and uncertainty. Calculating for instance analytically or numerically the fuzzy variance could be advantageous. Yet, this task is not simple, especially regarding the difficulties in measuring the multiplication of two fuzzy sets. These computational problems are not evident to overcome. Therefore, an approximation of this product is needed. In the aim of computing the fuzzy variance, we propose different approximations of this product including the one using a particular method called the signed distance. However, using some of our approximations, another computational complexity arises since we get non-positive fuzzy numbers due to the difference between two fuzzy sets. This implies a result given by a fuzzy number violating the principles of the n-uples notations. In order to solve this problem, we use the shifting (translation) techniques. In addition, we give a comparison between these approximations in the purpose of displaying their characteristics. Finally, we illustrate our approach by numerical examples. We highlight that one should be prudent when choosing an estimation of the fuzzy variance.

Since its inception in 1965, the theory of fuzzy sets has advanced in a variety of ways and in many disciplines. Applications of this theory can be found, for example, in artificial intelligence, computer science, medicine, control engineering, decision theory, expert systems, logic, management science, operations research, pattern recognition, and robotics. Mathematical developments have advanced to a very high standard and are still forthcoming to day. In this review, the basic mathematical framework of fuzzy set theory will be described, as well as the most important applications of this theory to other theories and techniques. Since 1992 fuzzy set theory, the theory of neural nets and the area of evolutionary programming have become known under the name of ‘computational intelligence’ or ‘soft computing’. The relationship between these areas has naturally become particularly close. In this review, however, we will focus primarily on fuzzy set theory. Applications of fuzzy set theory to real problems are abound. Some references will be given. To describe even a part of them would certainly exceed the scope of this review. Copyright © 2010 John Wiley & Sons, Inc.
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In testing statistical hypotheses, as in other statistical problems, we may be confronted with fuzzy concepts. This paper
deals with the problem of testing hypotheses, when the hypotheses are fuzzy and the data are crisp. We first introduce the
notion of fuzzy p-value, by applying the extension principle and then present an approach for testing fuzzy hypotheses by comparing a fuzzy
p-value and a fuzzy significance level, based on a comparison of two fuzzy sets. Numerical examples are also provided to illustrate
the approach.

We extend the classical approach of hypothesis testing to the fuzzy environment. We propose a method based on fuzziness of data and on fuzziness of hypotheses at the same time. The fuzzy p-value with its α-cuts is provided and we show how to defuzzify it by the signed distance method. We illustrate our method by numerical applications where we treat a one and a two sided test. For the one-sided test, applying our method to the same data and performing tests on the same significance level, we compare the defuzzified p-values between different cases of null and alternative hypotheses.

Preliminary review / Publisher's description: Statistical data are not always precise numbers, or vectors, or categories. Real data are frequently what is called fuzzy. Examples where this fuzziness is obvious are quality of life data, environmental, biological, medical, sociological and economics data. Also the results of measurements can be best described by using fuzzy numbers and fuzzy vectors respectively. Statistical analysis methods have to be adapted for the analysis of fuzzy data. In this book, the foundations of the description of fuzzy data are explained, including methods on how to obtain the characterizing function of fuzzy measurement results. Furthermore, statistical methods are then generalized to the analysis of fuzzy data and fuzzy a-priori information. Key Features: * Provides basic methods for the mathematical description of fuzzy data, as well as statistical methods that can be used to analyze fuzzy data. * Describes methods of increasing importance with applications in areas such as environmental statistics and social science. * Complements the theory with exercises and solutions and is illustrated throughout with diagrams and examples. * Explores areas such quantitative description of data uncertainty and mathematical description of fuzzy data. This work is aimed at statisticians working with fuzzy logic, engineering statisticians, finance researchers, and environmental statisticians. It is written for readers who are familiar with elementary stochastic models and basic statistical methods.

In a recent paper we have discussed certain general principles underlying the determination of the most efficient tests of statistical hypotheses, but the method of approach did not involve any detailed consideration of the question of a priori probability. We propose now to consider more fully the bearing of the earlier results on this question and in particular to discuss what statements of value to the statistician in reaching his final judgment can be made from an analysis of observed data, which would not be modified by any change in the probabilities a priori. In dealing with the problem of statistical estimation, R. A. Fisher has shown how, under certain conditions, what may be described as rules of behaviour can be employed which will lead to results independent of these probabilities; in this connection he has discussed the important conception of what he terms fiducial limits. But the testing of statistical hypotheses cannot be treated as a problem in estimation, and it is necessary to discuss afresh in what sense tests can be employed which are independent of a priori probability laws.(Received May 31 1933)(Accepted October 30 1933)

A definition of fuzzy test for testing statistical hypotheses with vague data is proposed. Then the general method for the construction of fuzzy tests for hypotheses concerning an unknown parameter against one-sided or two-sided alternative hypotheses is shown. This fuzzy test, contrary to the classical approach, leads not to the binary decision: to reject or to accept given null hypothesis, but to a fuzzy decision showing a grade of acceptability of the null and the alternative hypothesis, respectively. However, it is a natural generalization of the traditional test, i.e. if the data are precise, not vague, we get a classical statistical test with the binary decision. A measure of fuzziness of the considered fuzzy test is suggested and the robustness of that test is also discussed.

By using the decomposition principle and the crisp ranking system on R, we construct a new ranking system for fuzzy numbers which is very realistic and also matching our intuition as in R.

A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.

Since traditional sampling survey via questionnaire is difficult in reflecting interviewee's incomplete assessment and uncertain thought, we use fuzzy sense of sampling to express the degree of interviewee's feelings, and find that the result is closer to interviewee's real thought. In this study, we propose two algorithms to do aggregative assessment for sampling survey by signed distance method with the linear order character of symmetric fuzzy linguistics instead of using previous centroid method. As the result that if the membership function of the triangular fuzzy number is not an isosceles triangle, then, based on the maximum membership grade principle, to defuzzify triangular fuzzy number by the signed distance is better than by the centroid method. The proposed fuzzy assessment method on sampling survey analysis is easily to assess the sampling survey and make the aggregative evaluation. Since the proposed model in this study is to measure the group evaluation, the final value is more objective than just one evaluator's assessment. Moreover, if there is only one evaluator existing, the proposed model is also appropriate to assess.

The problem of defuzzification of a fuzzy test for testing statistical hypotheses with vague data is considered. Two defuzzification operators are discussed. It is shown that the superposition of a fuzzy test and these operators leads to decision strategies equivalent to the classical crisp tests. Moreover, another source of randomization in hypothesis testing is indicated, and a new ground for the randomized tests application is given. Advice to practitioners – how to defuzzify – is also given.

Traditional sampling survey via questionnaire is difficult in reflecting interviewee's incomplete assessment and uncertain thought. Therefore, if we can use fuzzy sense of sampling to express the degree of interviewee's feelings based on his own concept, the result will be closer to interviewee's real thought. In this study, a new method for the defuzzification of fuzzy assessment for sampling survey is developed in this paper. We propose a model to do aggregative assessment for sampling survey by signed distance method with the linear order character of symmetric fuzzy linguistics. The proposed fuzzy assessment method on sampling survey analysis is easily to assess the sampling survey and make the aggregative evaluation.

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.