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... Gil and Casals [13] used the likelihood ratio in a hypothesis testing procedure where fuzziness is contained in the data. In Berkachy and Donzé [5], we proposed a practical procedure to construct confidence intervals by the likelihood ratio method which is seen in some sense general. The SGD Metric". ...

... We presented in Berkachy and Donzé [7] and Berkachy and Donzé [5] a generalisation of the traditional construction procedure. The aim was to show a practical tool based on the concept of likelihood ratio method to estimate fuzzy confidence intervals, in which the fuzziness contained in the variables is conveniently taken into consideration. ...

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.

... Finally, likewise for the non-fuzzy case, inference on and can be performed using maximum-likelihood (ML) theory (Denoeux 2011;McLachlan and Peel 2004) and, consequently, hypothesis testing on model's parameters can be performed using fuzzy version of likelihood ratio test (e.g., see Berkachy and Donzé 2019;Najafi et al. 2010). In this context, as for the maximum-likelihood theory under the EM procedure, inferential results are based on the asymptotic properties of ML-based estimators. ...

... To account for heterogeneity in the response variable rdb, two further models were estimated, one which included the dichotomous variable sex, and the other which also included driving_frequency. In order to evaluate models improvements, both the models were compared in terms of fuzzy likelihood-ratio test (Berkachy and Donzé 2019). Table 4 reports the results for model 2 and model 3. ...

Modeling human ratings data subject to raters’ decision uncertainty is an attrac-tive problem in applied statistics. In view of the complex interplay between emo-tion and decision making in rating processes, final raters’ choices seldom reflect the true underlying raters’ responses. Rather, they are imprecisely observed in the sense that they are subject to a non-random component of uncertainty, namely the deci-sion uncertainty. The purpose of this article is to illustrate a statistical approach to analyse ratings data which integrates both random and non-random components of the rating process. In particular, beta fuzzy numbers are used to model raters’ non-random decision uncertainty and a variable dispersion beta linear model is instead adopted to model the random counterpart of rating responses. The main idea is to quantify characteristics of latent and non-fuzzy rating responses by means of ran-dom observations subject to fuzziness. To do so, a fuzzy version of the Expecta-tion–Maximization algorithm is adopted to both estimate model’s parameters and compute their standard errors. Finally, the characteristics of the proposed fuzzy beta model are investigated by means of a simulation study as well as two case studies from behavioral and social contexts.

... , r n against the indices of the observations in order to check for particular trends or patterns in the predicted data. Finally, likewise for the non-fuzzy case, inference on β and γ can be performed using maximum-likelihood theory [24,51] and, consequently, hypothesis testing on model's parameters can be performed using fuzzy version of likelihood ratio test (e.g., see [7,57]). ...

... To account for heterogeneity in the response variable rdb, two further models were estimated, one which included the dichotomous variable sex, and the other which also included driving_frequency. In order to evaluate models improvements, both the models were compared in terms of fuzzy likelihood-ratio test [7]. Table 4 reports the results for model 2 and model 3. ...

Modeling human ratings data subject to raters' decision uncertainty is an attractive problem in applied statistics. In view of the complex interplay between emotion and decision making in rating processes, final raters' choices seldom reflect the true underlying raters' responses. Rather, they are imprecisely observed in the sense that they are subject to a non-random component of uncertainty, namely the decision uncertainty. The purpose of this article is to illustrate a statistical approach to analyse ratings data which integrates both random and non-random components of the rating process. In particular, beta fuzzy numbers are used to model raters' non-random decision uncertainty and a variable dispersion beta linear model is instead adopted to model the random counterpart of rating responses. The main idea is to quantify characteristics of latent and non-fuzzy rating responses by means of random observations subject to fuzziness. To do so, a fuzzy version of the Expectation-Maximization algorithm is adopted to both estimate model's parameters and compute their standard errors. Finally, the characteristics of the proposed fuzzy beta model are investigated by means of a simulation study as well as two case studies from behavioral and social contexts.

... Alternatively, they can also be obtained by means of non-parametric or parametric bootstrap techniques [56]. Finally, it should be remarked that inference about ρ jk can be made based on the asymptotic results of fuzzy likelihood ratio statistics (e.g., see [57]). ...

This research concerns the estimation of latent linear or polychoric correlations from fuzzy frequency tables. Fuzzy counts are of particular interest to many disciplines including social and behavioral sciences, and are especially relevant when observed data are classified using fuzzy categories - as for socio-economic studies, clinical evaluations, content analysis, inter-rater reliability analysis - or when imprecise observations are classified into either precise or imprecise categories - as for the analysis of ratings data or fuzzy coded variables. In these cases, the space of count matrices is no longer defined over naturals and, consequently, the polychoric estimator cannot be used to accurately estimate latent linear correlations. The aim of this contribution is twofold. First, we illustrate a computational procedure based on generalized natural numbers for computing fuzzy frequencies. Second, we reformulate the problem of estimating latent linear correlations from fuzzy counts in the context of Expectation-Maximization based maximum likelihood estimation. A simulation study and two applications are used to investigate the characteristics of the proposed method. Overall, the results show that the fuzzy EM-based polychoric estimator is more efficient to deal with imprecise count data as opposed to standard polychoric estimators that may be used in this context.

This chapter first shows the definition of a fuzzy hypothesis. We after display the construction of a given fuzzy confidence interval. One of the highlights of this chapter is a new procedure of construction of fuzzy confidence intervals by the likelihood ratio method using the bootstrap technique. Moreover, we show in detail the hypotheses testing approaches, based on these intervals, followed by the fuzzy p-values. Both the fuzzy decisions and the fuzzy p-values are afterward defuzzified. We propose to defuzzify them by the signed distance operator from one side and from the generalized signed distance given from another one. The purpose is to sort out the main differences and drawbacks that might occur when using both distances in such contexts. All these procedures are illustrated by multiple detailed examples. Applications on a financial data set are also provided. In addition, a discussion on the comparison between the classical and fuzzy approaches is given. This chapter is closed by some guidelines on the use of each one of both approaches. The choice between the classical and fuzzy hypotheses testing approaches should be well-argued.

A fuzzy test for testing statistical hypotheses about an imprecise parameter is proposed for the case when the available data are also imprecise. The proposed method is based on the relationship between the acceptance region of statistical tests at level ? and confidence intervals for the parameter of interest at confidence level 1 ? ?. First, a fuzzy confidence interval is constructed for the fuzzy parameter of interest. Then, using such a fuzzy confidence interval, a fuzzy test function is constructed. The obtained fuzzy
test, contrary to the classical approach, leads not to a binary decision (i.e. to reject or to accept the given null hypothesis) but to a fuzzy decision showing the degrees of acceptability of the null and alternative hypotheses. Numerical examples are given to demonstrate the theoretical results, and show the
possible applications in testing hypotheses based on fuzzy observations.

In the present paper we are going to extend the likelihood ratio test to the case in which the available experimental information
involves fuzzy imprecision (more precisely, the observable events associated with the random experiment concerning the test
may be characterized as fuzzy subsets of the sample space, as intended by Zadeh, 1965). In addition, we will approximate the
immediate intractable extension, which is based on Zadeh’s probabilistic definition, by using the minimum inaccuracy principle
of estimation from fuzzy data, that has been introduced in previous papers as an operative extension of the maximum likelihood
method.

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.

Even though classical point and interval estimations (PIE) are one of the most studied fields in statistics, there are a few numbers of studies covering fuzzy point and interval estimations. In this pursuit, this study focuses on analyzing the works on fuzzy PIE for the years between 1980 and 2015. In the chapter, the literature is reviewed through Scopus database and the review results are given by graphical illustrations. We also use the extensions of fuzzy sets such as interval-valued intuitionistic fuzzy sets (IVIFS) and hesitant fuzzy sets (HFS) to develop the confidence intervals based on these sets. The chapter also includes numerical examples to increase the understandability of the proposed approaches.

In information processing tasks, sets may have a conjunctive or a disjunctive reading. In the conjunctive reading, a set represents an object of interest and its elements are subparts of the object, forming a composite description. In the disjunctive reading, a set contains mutually exclusive elements and refers to the representation of incomplete knowledge. It does not model an actual object or quantity, but partial information about an underlying object or a precise quantity. This distinction between what we call ontic vs. epistemic sets remains valid for fuzzy sets, whose membership functions, in the disjunctive reading are possibility distributions, over deterministic or random values. This paper examines the impact of this distinction in statistics. We show its importance because there is a risk of misusing basic notions and tools, such as conditioning, distance between sets, variance, regression, etc. when data are set-valued. We discuss several examples where the ontic and epistemic points of view yield different approaches to these concepts.

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.

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.

A method is proposed for estimating the parameters in a parametric statistical model when the observations are fuzzy and are assumed to be related to underlying crisp realizations of a random sample. This method is based on maximizing the observed-data likelihood defined as the probability of the fuzzy data. It is shown that the EM algorithm may be used for that purpose, which makes it possible to solve a wide range of statistical problems involving fuzzy data. This approach, called the fuzzy EM (FEM) method, is illustrated using three classical problems: normal mean and variance estimation from a fuzzy sample, multiple linear regression with crisp inputs and fuzzy outputs, and univariate finite normal mixture estimation from fuzzy data.

We extend the notion of confidence region to fuzzy data, by defining a pair of fuzzy inner and outer confidence regions. We show the connection with previous proposals, as well as with recent studies on hypothesis testing with low quality data.

- J Gebhardt
- M A Gil
- R Kruse

J. Gebhardt, M. A. Gil, R. Kruse, Fuzzy Set-Theoretic Methods in Statistics, in Fuzzy Sets
in Decision Analysis, Operations Research and
Statistics, Springer US, Boston, MA, 1998, pp.
311-347.

Fuzzy Confidence Regions, in Fuzzy Statistical Decision-Making: Theory and Applications

- R Viertl
- S M Yeganeh

R. Viertl, S. M. Yeganeh, Fuzzy Confidence Regions, in Fuzzy Statistical Decision-Making: Theory and Applications, Springer International Publishing, Cham, 2016, pp. 119-127.