Chapter

# Fuzzy Confidence Regions

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## Abstract

Confidence regions are usually based on exact data. However, continuous data are always more or less non-precise, also called fuzzy. For fuzzy data the concept of confidence regions has to be generalized. This is possible and the resulting confidence regions are fuzzy subsets of the parameter space. The construction is explained for classical statistics as well as for Bayesian analysis. An example is given in the last section.

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... Several researchers have afterwards proposed refined definitions of fuzzy confidence intervals. For instance, Viertl and Yeganeh [22] proposed a definition of the so-called confidence regions. Their main application was in the Bayesian context. ...
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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.
Chapter
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.
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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.