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There have been changes since the companies have realized the important role of quality improvement in their success. If they are able to produce high quality products and satisfy demands, then they can survive in competitive global markets. Quality improvement applications aim to decrease variability, which leads to less cost, production time, number of defects, scrap, rework and more customer satisfaction. Quality can be improved by reducing product variability. On the other hand, uncertainty or subjectivity is a part of many engineering and real life problems. However, these problems cannot be solved by traditional methods. This study focuses on constructing Xbar and S control charts in fuzzy environment. The approach is developed by considering the theoretical structure of the Shewhart control charts. The core of the approach depends on the combination of parametric interval estimation and fuzzy statistics. Control limits and samples are presented by fuzzy numbers which ensures to maintain fuzziness in control charts. An important property of the approach is that the fuzzy charts can be reduced to Shewhart control charts. A simulation study was conducted for the performance evaluation of fuzzy Xbar and S control charts. The proposed fuzzy control chart is sensitive to process mean shifts and variance changes, and outperforms the traditional control charts under the changes of variance. In addition, an example from the literature shows that the approach is an effective way of presenting fuzziness in the quality characteristics, which enables the approach to have high applicability to the real life problems.

In this paper, we develop fuzzy, possibilistic hypothesis tests for testing crisp hypotheses for a distribution parameter from crisp data. In these tests, fuzzy statistics are used, which are produced by the possibility distribution of the estimated parameter, constructed by the known from crisp statistics confidence intervals. The results of these tests are in much better agreement with crisp statistics than the ones produced by the respective tests of a popular book on fuzzy statistics, which uses fuzzy critical values. We also present an error that we found in the implementation of the unbiased fuzzy estimator of the variance in this book, due to a poor interpretation of its mathematical content, which leads to disagreement of some fuzzy hypotheses tests with their respective crisp ones. Implementing correctly this estimator, we produce test statistics that achieve results in hypotheses tests that are in much better agreement with the results of the respective crisp ones.

Uncertainty makes the risk evaluation of complex water transportation systems (WTSs) a difficult task. To achieve reasonable results while accounting for uncertainty, the risk evaluation of nautical navigational environments (NNEts) is often based on classical cloud model theory. This study proposes the concept of a risk cloud model (RCM) for NNEt evaluation and uses a fuzzy statistics-based computational approach to obtain the RCM parameters. As a case study, the proposed RCM method was applied to the risk evaluation of the Qiongzhou Strait. The performance of the proposed method was compared to those of a fuzzy theory-based method and an earlier proposed simplified algorithm. The results of the case study demonstrated the effectiveness of the proposed method along with several key advantages. First, the method could deal with uncertainty, take advantage of multichannel information, and evaluate risk features. Second, the RCM droplets intuitively displayed the qualitative and quantitative characteristics of risk levels, which facilitated understanding and analysis. Third, it showed a good sensitivity to ensure the refinement of evaluation results. The proposed method offered an improved approach to NNEt risk evaluation under uncertain conditions.

The aim of the paper is to examine procedures of descriptive statistics in the case when the values of relevant attribute in a sample are set in the form of fuzzy categories. The paper provides alternative definitions of a fuzzy random variable, and describes corresponding procedures for calculating the analogues of location and spread parameters. The paper also presents some illustrative examples and analyses the results obtained. Based on the result analysis, practical recommendations are given on how to use procedures of fuzzy statistics.

In this paper, we study the problem of defining statistical parameters when the uncertainty is expressed using a fuzzy measure. We extend the concept of monotone expectation in order to define a monotone variance and monotone moments. We also study parameters that allow the joint analysis of two functions defined over the same reference set. Finally, we propose some parameters over product spaces, considering the case in which a function over the product space is available and also the case in which such function is obtained by combining those in the marginal spaces.

We shall consider fuzzy hypotheses test for signed-rank Wilcoxon fuzzy test by fuzzy difference on rate of internal division. Fundamental to these discussion are fuzzy number data and Wilcoxon signed-rank fuzzy test of a fuzzy hypothesis H f0 which is based upon a fuzzy statistics whose distribution does not depend upon the specified distribution or any parameters.

The purpose of this study is to compare the reliability of Likert scale between crisp and fuzzy data. The survey data is simulated based on two kinds of questionnaire data. They are questionnaire of crisp data and fuzzy data respectively. According to the viewpoints of fuzzy logic, human thinking is multi-value and fuzzy data will be more appropriate for survey. Therefore, it is proposed that the reliability from fuzzy data will be higher. Results of the simulation show that reliability of fuzzy data performs better than crisp data. Based on the findings of this study, some suggestions and recommendations are discussed for future research.

In this paper we present a crisp-input/fuzzy-output regression model based on the rationale of generalized maximum entropy (GME) method. The approach can be used in several situations in which one have to handle with particular problems, such as small samples, ill-posed design matrix (e.g., due to the multicollinearity), estimation problems with inequality constraints, etc. After having described the GME-fuzzy regression model, we consider an economic case study in which the features provided from GME approach are evaluated. Moreover, we also perform a sensitivity analysis on the main results of the case study in order to better evaluate some features of the model. Finally, some critical points are discussed together with suggestions for further works.

In this paper, on the basis of Zadeh’s probability measure of fuzzy events, the p-value concept is generalized for testing fuzzy hypotheses. We prove that the introduced p-value has uniform distribution over (0, 1) when the null fuzzy hypothesis is true. Then, based on such a p-value, a procedure is illustrated to test various types of fuzzy hypotheses. Several applied examples are given to show the performance of the method.

Fuzzy modeling and fuzzy statistics provide useful tools for handling empirical situations affected by vagueness and imprecision in the data. Several fuzzy statistical models and methods (e.g., fuzzy regression, fuzzy principal component analysis, fuzzy clustering) have been developed over the years. Generally the standard LR-fuzzy data representation has been used in these methods. However, several empirical contexts, such as human ratings and decision making, may show more complex fuzzy structures which cannot be successfully modeled by the LR representation. In all these cases another type of fuzzy data representation, the so-called LHIR representation, should be preferred instead. In particular, this novel representation allows to handle with fuzzy data which are characterized by non-convex membership functions. In this paper, we address the problem of summarizing large datasets characterized by two-mode non-convex fuzzy data. We introduce a novel dimension reduction technique (NCFCA) based on the framework of Component Analysis and Least squares programming. Finally, to better highlight some important characteristics of the proposed model, we apply NCFCA to three empirical datasets concerning behavioral and socio-economic issues.

Graphical abstractDisplay Omitted HighlightsWe consider two fuzzy regression models from fuzzy least squares tradition.We rewrite these models within the Generalized Maximum Entropy Approach of estimation.We compare LS and GME approaches in the multicollinearity problem.Monte Carlo studies show increasing multicollinearity GME outperforms LS in efficiency.Empirical evidence shows some applicative advantages of GME. Fuzzy statistics provides useful techniques for handling real situations which are affected by vagueness and imprecision. Several fuzzy statistical techniques (e.g., fuzzy regression, fuzzy principal component analysis, fuzzy clustering) have been developed over the years. Among these, fuzzy regression can be considered an important tool for modeling the relation between a dependent variable and a set of independent variables in order to evaluate how the independent variables explain the empirical data which are modeled through the regression system. In general, the standard fuzzy least squares method has been used in these situations. However, several applicative contexts, such as for example, analysis with small samples and short and fat matrices, violation of distributional assumptions, matrices affected by multicollinearity (ill-posed problems), may show more complex situations which cannot successfully be solved by the fuzzy least squares. In all these cases, different estimation methods should instead be preferred. In this paper we address the problem of estimating fuzzy regression models characterized by ill-posed features. We introduce a novel fuzzy regression framework based on the Generalized Maximum Entropy (GME) estimation method. Finally, in order to better highlight some characteristics of the proposed method, we perform two Monte Carlo experiments and we analyze a real case study.

How to create vague environment is the precondition of the application based on vague set in recommender systems of electric commerce and so on. According to fuzzy statistics, fuzzy set data of commodity attributes are easy to get to. So, it is worth to research on the method of transforming Fuzzy set into Vague set. This paper presents a new method of transforming Fuzzy set into Vague set, and apply its in recommender systems of electric commerce based on content. The experimental results show that this method is effective and practical, which can not only improve the accuracy of commodity recommended but also provide a new idea of the application of vague set.

The notion of first- and second-order fuzzy statistics of digital images is presented. Owing to the inherent imprecision in the gray values, fuzzy statistics have been observed to behave better in representing the spatial gray distribution in a digital image.

We have two populations: Pop I and Pop II. Pop I is normally distributed with unknown mean μ 1 and unknown variance σ 1 2 . Pop II is also normally distributed with unknown mean μ 2 and unknown variance σ 2 2 . We wish to do the following statistical test $$\mathop H\nolimits_0 :\mathop \mu \nolimits_1 - \mathop \mu \nolimits_1 = 0$$ (17.1) verses $$\mathop H\nolimits_0 :\mathop \mu \nolimits_1 - \mathop \mu \nolimits_1 \ne 0$$ (17.2) We collect a random sample of size n 1 from Pop I and let \({{\bar{x}}_{1}}\) be the mean for this data and s 1 2 is the sample variance. We also gather a random sample of size n 2 from Pop II and \({{\bar{x}}_{2}}\) is the mean for the second sample with s 2 2 the variance. We assume these two random samples are independent.

A method of classification of weather forecasts is approached by applying fuzzy grade statistical theory. This method has been verified with an example, and the correct rate for the method is 87%.

Fuzzy set theory models situations in which the uncertainty is due to the non-precise (fuzzy) environment. One such case is testing the hypotheses problem where hypotheses are fuzzy rather than crisp and the data are crisp. Pais and Benton (1997) [1] present a suitable amount of cadmium absorption which is more appropriate for modeling by a fuzzy subset. In this paper, on real-world agricultural data generated by Ivani (2007) [2] , the suitableness of the mean absorption cadmium in the plant from the polluted soil is tested at the given significance level. Due to uncertainty in the suitable amount of cadmium absorption, in order to test this amount, we use fuzzy hypotheses testing instead of classical hypotheses testing. The fuzzy p -value approach is used for this test to conclude whether or not the mean absorption of cadmium coincides with the proposed amounts by Pais and Benton [1] . As expected, in fuzzy hypotheses testing, the degree of acceptance or rejection of the null fuzzy hypothesis is computed for each treatment of pollution. The data are from two plants: radish and cress, which are experimented on in soil polluted with CdNO3 salt. The results showed that using classical hypotheses testing may lead to contradictory decisions, and the proposed fuzzy hypotheses testing is a rational substitute for classical hypotheses testing when the environment is in a vague status.

In this paper we shall discuss some problems concerning the fuzzy statistics and the random subsets. The Rain space, a framework treating the fuzzy subsets as random subsets will be introduced. In this space, the measurability of random subsets is equivalent to the strong measurability. Applying the graph of random subsets we will give a correspondence theorem which combines the falling random subsets and the measurable fuzzy subsets. Finally, we will discuss the operations of fuzzy subsets, some interesting results will be found.