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In this paper, a nonlinear time series model is developed for the case when the underlying time series data are reported by LR fuzzy numbers. To this end, we present a three-stage nonparametric kernel-based estimation procedure for the center as well as the left and right spreads of the unknown nonlinear fuzzy smooth function. In each stage, the nonparametric Nadaraya–Watson estimator is used to evaluate the center and the spreads of the fuzzy smooth function. A hybrid algorithm is proposed to estimate the unknown optimal bandwidths and autoregressive order simultaneously. Various goodness-of-fit measures are utilized for performance assessment of the fuzzy nonlinear kernel-based time series model and for comparative analysis. The practical applicability and superiority of the novel approach in comparison with further fuzzy time series models are demonstrated via a simulation study and some real-life applications.

In profile monitoring, it is usually assumed that the observations between or within each profile are independent of each other. However, this assumption is often violated in manufacturing practice, and it is of utmost importance to carefully consider autocorrelation effects in the underlying models for profile monitoring. For this reason, various statistical control charts have been proposed to monitor profiles when between- or within-data is correlated in Phase II, in which the main aim is to develop control charts with quicker detection ability. As a novel approach, this study aims to employ machine learning techniques as control charts instead of statistical approaches in monitoring profiles with between-profile autocorrelations. Specifically, new input features based on conventional statistical control chart statistics and normalized estimated parameters are defined that are capable of adequately accounting for the between-autocorrelation effect of profiles. In addition, six machine learning techniques are extended and compared by means of Monte Carlo simulations. The simulation results indicate that machine learning techniques can obtain more accurate results compared with statistical control charts. Moreover, adaptive neuro-fuzzy inference systems outperform other machine learning techniques and the conventional statistical control charts.

The sign test is one of the most popular nonparametric tests for location problems and allows testing for any quantile of a population. However, the common sign test has serious drawbacks such as loss of information by considering solely signs of observations but not their magnitudes, various problems related to handling of ties in the data, and the lack of embedding uncertainty regarding the fraction of underlying quantile. To address these issues, we present an extended sign test based on fuzzy categories and fuzzy formulated hypotheses that improves the generality, versatility, and practicability of the common sign test. This generalized test procedure is neat in theory and practice and avoids disadvantages that are often associated with fuzzy tests (e.g., a considerably higher complexity of the underlying model, a fuzzy test decision, and a possibilistic instead of a probabilistic interpretation of test results). In addition, we perform a comprehensive case study on COVID‐19 in HIV‐infected individuals with a focus on human body temperature and related measurement problems. The results of the study clearly indicate that fuzzy categories and fuzzy hypotheses improve the performance of the sign test.

Although it is well known that the performance of attribute control charts decreases significantly when the assumption of known process parameters is invalid, this assumption is prevalent in the pertinent literature. However, in most practical applications, the process parameters have to be estimated from a finite in-control Phase I sample, and therefore the performance of attribute control charts should be evaluated from the perspective of estimated process parameters. In this paper, we compare the run length properties of the hypergeometric n p chart in both the known and estimated parameter cases. In particular, we investigate the required number of Phase I samples and new specific chart parameters that allow the hypergeometric n p chart with estimated parameter p to have approximately the same in-control performance as in the known parameters case. Moreover, we perform a comprehensive in-control and out-of-control comparison of the hypergeometric n p chart with its binomial counterpart. In order to achieve these objectives, we also present a new approach to effectively compute the probability distribution of the sum of independent and identically hypergeometric-distributed random variables. The proposed approximation reduces the computational effort to a few seconds while keeping a remarkable high accuracy with only negligible deviations compared to the exact distribution obtained via convolution.

International audience; In this paper, we present a number-between-events (NBE) control chart for monitoring the fraction nonconforming in nite horizon production (FHP) processes and related specic performance measures. When monitoring fractions nonconforming in FHP processes, the common binomial p-chart has two crucial limitations: the underlying distributional assumptions are violated when dealing with low-volume production and a scarce eciency in the case of processes characterized by a low fraction nonconforming. Thus, an ecient monitoring of FHP processes requires the selection of the correct underlying statistical model: in this case, a distribution from the hypergeometric family of discrete statistical distributions. An ecient statistical monitoring of processes with low fractions nonconforming can be achieved by means of discrete time-between-events (TBE) control charts, which count the number of units up to the appearance of a xed number of nonconforming units in the sample. Here, we present a discrete TBE-chart, denoted as NBE-chart, based on the negative hypergeometric distribution that meets numerous requirements of ecient monitoring of FHP processes. The proposed control chart can be conveniently used for both low-volume and mass production in processes with frequent changeovers.

Profile monitoring is a challenging issue in statistical process control (SPC). It aims to use a functional relationship between a response variable and one or more explanatory variable(s) to summarize the quality of a process/product. Most of the existing studies consider the same form of a functional relationship for both in-control (IC) and out-of-control (OC) situations or parametric approaches. However, non-parametric profiles with different relationships in OC conditions are very common. In this paper, we propose a novel ensemble framework to monitor changes in both regression relationship and variation of the profile for Phase II applications. This approach employs a pool of artificial neural networks (ANNs) as learners to enhance the efficiency of a base control chart, which is a non-parametric exponentially weighted moving average (NEWMA) in this study. Then, a further ANN is used as a reasoning scheme (incorporator) to perform prediction by combining the outcomes of the learners. Experimental results demonstrate the effectiveness of the proposed framework, denoted by EANNN, in comparison with the base control chart, i.e., NEWMA, and other non-parametric methods. In addition, a practical example regarding a deep reactive ion-etching process from semiconductor device fabrication is provided to show the implementation of the proposed method. 2022 Elsevier Ltd Scopus

Nonparametric tests do not rely on data belonging to any particular parametric family of probability distributions, which makes them preferable in case of doubt about the underlying population. Although the two-tailed sign test is likely the most common nonparametric test for location problems, practitioners face serious drawbacks, such as its lack of statistical power and its inapplicability when information regarding data and hypotheses is uncertain or imprecise. In this paper, we generalize the two-tailed sign test by embedding fuzzy hypotheses caused by uncertainty/imprecision regarding linguistic statements on fractions of underlying quantiles. By achieving this objective, (1) crucial limitations of the common two-tailed sign test are mitigated/overcome, (2) various further strengths are incorporated into the sign test (e.g., meeting the trade-off between point- and interval-valued hypotheses, facilitated formulation of fuzzy hypotheses, standardization of membership functions), and (3) shortcomings that often come along with fuzzy hypothesis testing are avoided (e.g., higher complexity, fuzzy test decision, possibilistic interpretation of test results). In addition, we conduct a comprehensive case study using a real data set on the psychosocial status during the COVID-19 pandemic. The results of the case study clearly indicate that the generalized two-tailed sign test is preferable to the two-tailed sign test with point- or interval-valued hypotheses.

Control charts for monitoring fractions non-conforming are widely used in healthcare surveillance, industrial quality control, and service operations management. They are usually based on the assumption of Binomial-distributed variables, control limits approximated by the Normal distribution and infinitely large population size. However, there are many practical situations where the assumption of Binomial-distributed data is not satisfied, for instance when the underlying periodic population is of a finite size and random sampling complies with sampling without replacement. Moreover, the Normal approximation is often inappropriate and efficient monitoring requires control charts to be able to take varying population sizes into account. To overcome these drawbacks of the traditional p-chart, we present in this paper the Hypergeometric p-chart with probability control limits, where fraction data is assumed to follow the Hypergeometric distribution and control limits are calculated using exact Hypergeometric probabilities in compliance with time-varying sample and population sizes. This approach enables a dynamic and thus more general modelling of the control limits, i.e. the probability control limits are capable of accommodating more complex situations in practice in a natural way.

PurposeIn acceptance sampling, the hypergeometric operating characteristic (OC) function (so called type-A OC) is used to be approximated by the binomial or Poisson OC function, which actually reduce computational effort, but do not provide suffcient approximation results. The purpose of this paper is to examine binomial- and Poisson-type approximations to the hypergeometric distribution, in order to find a simple but accurate approximation that can be successfully applied in acceptance sampling.Design/methodology/approachThe authors present a new binomial-type approximation for the type-A OC function, and derive its properties. Further, the authors compare this approximation via an extensive numerical study with other common approximations in terms of variation distance and relative efficiency under various conditions on the parameters including limiting cases.FindingsThe introduced approximation generates best numerical results over a wide range of parameter values, and ensures arithmetic simplicity of the binomial distribution and high accuracy to meet requirements regarding acceptance sampling problems. Additionally, it can considerably reduce the computational effort in relation to the type-A OC function and therefore is strongly recommended for calculating sampling plans.Originality/valueThe newly presented approximation provides a remarkably close fit to the type-A OC function, is discrete and needs no correction for continuity, and is skewed in the same direction by roughly the same amount as the exact OC. Due to less factorials, this OC in general involves lower powers than the type-A OC function. Moreover, the binomial-type approximation is easy to fit to the conventional statistical computing packages.

In this article, we propose the generalized one-tailed hypergeometric test with fuzzy hypotheses for both test methods: significance testing (by controlling the type I error) and hypothesis testing...

In this paper we introduce the control chart for fraction non-conforming based on the Hypergeometric distribution, and propose the exact Hypergeometric test for a population proportion to detect significant changes in the process quality level. In contrast to the common Binomial p-chart, which is especially suitable for sampling from continuous production processes, the Hypergeometric p-chart can be established also for periodical production processes, where sampled items successively produced during one period are not stochastically independent and the probability of a sampled item being non-conforming depends on the previous sampled item. Further, while the Binomial p-chart does not take the fluctuation of the production process into account, the Hypergeometric p-chart enables consideration of the population size of periodical production appropriately. The Hypergeometric p-chart leads in general to lower probabilities of the type I error, i.e. false rejection of the hypothesis of statistical control, to a considerably higher in-control ARL, and thus to a significantly lower frequency of false alarms. Moreover, it provides monitoring of a manufacturing process with a considerably lower inspection effort than its Binomial counterpart. Finally, the exact Hypergeometric test leads to more accurate test decisions in detecting process shifts compared with the common used approximate Binomial test.

Highlights • This article presents theoretical bases of three methods for determining the probability distribution of the sum of i.i.d. hypergeometric random variables: (1) direct convolution, (2) recursive algorithm by De Pril, (3) approximation. • We provide associated MATLAB codes (including context-specific customizations) for direct implementation of these methods and discuss technical aspects and essential details of the tweaks we have made. • A representative application example in SPM shows that the proposed approximation is considerably simpler in application than both other methods and it ensures a remarkable high accuracy of the results while reducing computational time considerably., Graphical abstract, In probability theory and statistics, the probability distribution of the sum of two or more independent and identically distributed (i.i.d.) random variables is the convolution of their individual distributions. While convoluting random variables following a binomial, geometric or Poisson distribution is a straightforward procedure, convoluting hypergeometric-distributed random variables is not. The problem is that there is no closed form solution for the probability mass function (p.m.f.) and cumulative distribution function (c.d.f.) of the sum of i.i.d. hypergeometric random variables. To overcome this problem, we propose an approximation for the distribution of the sum of i.i.d. hypergeometric random variables. In addition, we compare this approximation with two classical numerical methods, i.e., convolution and the recursive algorithm by De Pril, by means of an application in Statistical Process Monitoring (SPM). We provide MATLAB codes to implement these three methods for computing the probability distribution of the sum of i.i.d. hypergeometric random variables in an efficient way. The obtained results show that the proposed approximation has remarkable properties and may be helpful in all fields, where the problem of convoluting hypergeometric-distributed random variables occurs. Therefore, the approximation considered in this paper is well suited to make a change over established practices.•This article presents theoretical bases of three methods for determining the probability distribution of the sum of i.i.d. hypergeometric random variables: (1) direct convolution, (2) recursive algorithm by De Pril, (3) approximation.•We provide associated MATLAB codes (including context-specific customizations) for direct implementation of these methods and discuss technical aspects and essential details of the tweaks we have made.•A representative application example in SPM shows that the proposed approximation is considerably simpler in application than both other methods and it ensures a remarkable high accuracy of the results while reducing computational time considerably.

This paper is concerned with fuzzy hypothesis testing in the framework of the randomized and non-randomized hypergeometric test for a proportion. Moreover, we differentiate between a test of significance and an alternative test to control the type I error or both error types simultaneously. In contrast to classical (non-)randomized hypothesis testing, fuzzy hypothesis testing provides an additional gradual consideration of the indifference zone in compliance with expert opinion or user priorities. In particular, various types of hypotheses with user-specified membership functions can be formulated. Additionally, the proposed test methods are compared via a comprehensive case study, which demonstrates the high flexibility of fuzzy hypothesis testing in practical applications.

Purpose The purpose of this paper is to construct innovative exact and approximative sampling plans for acceptance sampling in statistical quality control. These sampling plans are determined for crisp and fuzzy formulation of quality limits, various lot sizes and common α- and β-levels. Design/methodology/approach The authors use generalized fuzzy hypothesis testing to determine sampling plans with fuzzified quality limits. This test method allows a consideration of the indifference zone related to expert opinion or user priorities. In addition to the exact sampling plans calculated with the hypergeometric operating characteristic function, the authors consider approximative sampling plans using a little known, but excellent operating characteristic function. Further, a comprehensive sensitivity analysis of calculated sampling plans is performed, in order to examine how the inspection effort depends on crisp and fuzzy formulation of quality limits, the lot size and specifications of the producer’s and consumer’s risks. Findings The results related the parametric sensitivity analysis of the calculated sampling plans and the conclusions regarding the approximation quality provide the user a comprehensive basis for a direct implementation of the sampling plans in practice. Originality/value The constructed sampling plans ensure the simultaneous control of producer’s and consumer’s risks with the smallest possible inspection effort on the one hand and a consideration of expert opinion or user priorities on the other hand.

Anwendungen von Zustandsraummodellen und des Kalman-Filters sind in der stochastischen Schadenreservierung vergleichsweise unterreprasentiert. Dies liegt meist in ihrer hohen Komplexitat aufgrund Matrizen-basierter Ansatze begrundet, die praktische Anwendungen teils deutlich erschweren. In diesem Artikel wird ein skalares Zustandsraummodell fur Schadenstande vorgestellt, das aufgrund seiner skalaren Struktur nicht nur eine einfachere Handhabbarkeit mit sich bringt, sondern auch anschauliche und praktisch relevante Interpretationen der Komponenten des Zustandsraummodells ermoglicht. Neben einer umfassenden Darstellung des skalaren Zustandsraummodells werden einige empirische Anwendungsbeispiele und Vergleiche mit popularen Schadenreservierungsverfahren vorgenommen, die die Starken des skalaren Zustandsraummodells in praktischen Anwendungen aufzeigen. Das skalare Zustandsraummodell ist dabei eine robustifizierte Erweiterung des verteilungsfreien Chain-Ladder-Verfahrens, in der angenommen wird, dass den Beobachtungen unbeobachtbare Zustande zugrunde liegen. Unter Verwendung der Kalman-Rekursionen fur die Prognose, Filterung und Glattung kann das gesamte zugrunde liegende Abwicklungsquadrat bestimmt werden. Zudem liegen mit den Filterungen/Glattungen praktische Werkzeuge zur Identifizierung und Robustifizierung von Ausreisern sowie zur Interpolation fehlender Werte im Abwicklungsdreieck vor. Damit wird uberdies auch die Problematik fehlender Werte im oberen Abwicklungsdreieck auf naturliche Weise behoben.

In statistical inference hypotheses related to different kinds of phenomena are formulated, and then data are collected and analyzed, which either confirm or falsify these hypotheses. Considering traditional statistics, in the underlying models hypotheses and sample data should be well defined. However, these models are often inadequate with regard to real-life problems, as theoretical specifications and observed information are frequently imprecise, vague, incomplete, qualitative, linguistic or noisy. To relax this rigidity, numerous researchers have proposed modifications and extensions of statistical inference approaches with the help of concepts of fuzzy statistics. In the meantime there are many papers on the topic of hypothesis testing in fuzzy environments, especially based on fuzzy hypotheses and/or by using fuzzy data. In order to structure this variety of contributions, proposals and applications, we give a comprehensive systematic review in this paper and offer a bibliography on fuzzy hypothesis testing. The paper seeks to consolidate the topic of fuzzy hypothesis testing with the purpose of supporting new researchers in this field and highlighting potential directions for future research.

Statistical control charts have found valuable applications in health care, having been largely adopted from operations research in manufacturing. However, the most common types are not best-suited to monitor high-yield processes (outcomes comprising true/false fractions, ‘near-zero’) and periodical processes (characterized by sequences of single populations of finite sizes), but rather to monitor variable vital signs levels and, to a lesser degree, service performance indicators. We discuss control charts that are most suitable for fraction non-conforming measurements. We focus particularly on high-yield and periodical processes, i.e. range in which out-of-control conditions are expected and should be identified. For these conditions, we discuss control charts based on the family of hypergeometric distributions, explaining and comparing their application to more traditional alternatives with two health care case studies. We demonstrate that hypergeometric-type control charts provide higher sensitivity in timely identification of changing rare event fractions and are well-suited for monitoring of periodical processes, while remaining more resistant to false alarms, versus their alternatives.

Statistical regression analysis is a powerful and reliable method to determine the impact of one or several independent variable(s) on a dependent variable. It is the most widely used of all statistical methods and has broad applicability to numerous practical problems. However, various problems can arise, when for instance the sample size is too small, distributional assumptions are not fulfilled, the relationship between independent and dependent variables is vague or when there is an ambiguity of events. Moreover, the complexity of real-life problems often makes the underlying models inadequate, since information is frequently imprecise in many ways. To relax these rigidities, numerous researchers have modified and extended concepts of statistical regression analysis by means of concepts of fuzzy set theory. By now, there is a large number of papers on the topic of fuzzy regression analysis, especially concerning possibilistic, fuzzy least squares or machine learning approaches. Additionally, the variety of approaches includes probabilistic, logistic, type-2 and clusterwise fuzzy regression methods, among many others. Besides papers mainly devoted to advances in methodology, there are also several papers presenting case studies in various research fields. To structure this diversity of papers, proposals and applications we give in this paper a comprehensive systematic review and provide a bibliography on the topic of fuzzy regression analysis. Thus, the paper intends to consolidate the topic in order to aid new researchers in this area, focuses the field’s attention on key open questions, and highlights possible directions for future research.