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In this paper, a nonlinear Autoregressive Moving Average (ARMA) time series model is developed for the case where observations are affected by fuzziness. The primary motivation is to address the limitations of ARMA models, specifically their inability to effectively model complex nonlinear relationships, handle long memory processes, and manage non- Gaussian data. To achieve this, a fuzzy ARMA model is estimated using a method based on artificial neural networks. The objective is to construct a robust fuzzy time series model by employing various popular activation functions, such as logistic, hyperbolic tangent, and rectified linear unit. The effectiveness of the proposed model is rigorously evaluated using three well-established performance criteria. Furthermore, to demonstrate the practical benefits and applicability of this new time series model, a comparative analysis using both simulated data and real-world examples is conducted. [ABSTRACT FROM AUTHOR]

This paper proposes a robust economic-statistical design for Hotelling's T2 control chart to monitor linear profiles, accounting for uncertain parameters. Unlike previous studies assuming fixed inputs, this work applies an uncertainty interval approach to address parameter uncertainty when investigating the T2 chart design. In this way the challenge of parameter uncertainty's adverse impact on the effectiveness of quality control models in practice is addressed. The key contribution of this study is formulating a T2 control chart that maintains accuracy despite uncertainty. Results demonstrate the value of increased costs to mitigate uncertainty risks. The proposed approach also facilitates selecting appropriate protection levels against uncertainty. The paper examines practical challenges and applications of implementing the T2 chart under real-world conditions affected by uncertainty. Overall, the proposed robust T2 control chart design incorporates parameter ranges for monitoring linear profiles, contrasting with previous models assuming fixed parameters. [ABSTRACT FROM AUTHOR]

The sample strategy employed in statistical parameter estimation issues has a major impact on the accuracy of the parameter estimates. Ranked set sampling (RSS) is a highly helpful technique for gathering data when it is difficult or impossible to quantify the units in a population. A bounded power logarithmic distribution (PLD) has been proposed recently, and it may be used to describe many real-world bounded data sets. In the current work, the three parameters of the PLD are estimated using the RSS technique. A number of conventional estimators using maximum likelihood, minimum spacing absolute log-distance, minimum spacing square distance, Anderson-Darling, minimum spacing absolute distance, maximum product of spacings, least squares, Cramer-von-Mises, minimum spacing square log distance, and minimum spacing Linex distance are investigated. The different estimates via RSS are compared with their simple random sampling (SRS) counterparts. We found that the maximum product spacing estimate appears to be the best option based on our simulation results for the SRS and RSS data sets. Estimates generated from SRS data sets are less efficient than those derived from RSS data sets. The usefulness of the RSS estimators is also investigated by means of a real data example. [ABSTRACT FROM AUTHOR]

Statistical modeling of epidemiological curves to capture the course of epidemic processes and to implement a signaling system for detecting significant changes in the process is a challenging task, especially when the process is affected by political measures. As previous monitoring approaches are subject to various problems, we develop a practical and flexible tool that is well suited for monitoring epidemic processes under political measures. This tool enables monitoring across different epochs using a single statistical model that constantly adapts to the underlying process, and therefore allows both retrospective and on‐line monitoring of epidemic processes. It is able to detect essential shifts and to identify anomaly conditions in the epidemic process, and it provides decision‐makers a reliable method for rapidly learning from trends in the epidemiological curves. Moreover, it is a tool to evaluate the effectivity of political measures and to detect the transition from pandemic to endemic. This research is based on a comprehensive COVID‐19 study on infection rates under political measures in line with the reporting of the Robert Koch Institute covering the entire period of the pandemic in Germany. [ABSTRACT FROM AUTHOR]

Mathematical data science combines mathematical techniques with data science methods to extract insights from data. It involves working with data at all stages of the data life cycle and using various tools and techniques to analyze data. This field is applied in business, industry, and medicine to make predictions, identify patterns, and draw conclusions from data. The special issue of Mathematics presents ten papers that showcase innovative mathematical data science techniques in solving complex problems across different domains. These papers cover topics such as modeling failure times, time series analysis, process capability indices, control charts, internet usage monitoring, and more. The authors emphasize the importance of integrating advanced mathematical techniques with emerging technologies to drive innovation in data science. [Extracted from the article]

This document is an editorial introducing a special issue of the journal "Symmetry" on the topic of symmetrical and asymmetrical distributions in statistics and data science. The editorial provides a summary of the contributions to the special issue, which include studies on topics such as control charts, copulas, statistical distributions, discretization methods, spatial dependence, queuing models, survival analysis, and reliability analysis. The authors invite researchers to submit contributions to a second volume of the special issue, with a potential focus on the hypergeometric and negative hypergeometric distributions. [Extracted from the article]

Statistical inference is the process of drawing conclusions about underlying population(s) using sample data to either confirm or falsify hypotheses. However, the complexity of real-life problems often makes the underlying statistical models inadequate, as information is often imprecise in many respects. To address this common problem, some papers have been published on modifications and extensions of test concepts by employing tools of fuzzy statistics. In this paper, we present a non-parametric test for the difference between quantiles of two independent populations based on fuzzy random variables. For this purpose, we consider the fuzzy quantile function and its estimation based on α -values of fuzzy random variables. We then provide a fuzzy test based on the fuzzy empirical distribution function for the difference of fuzzy order statistics from these independent populations. We also suggest a specific degree-based criterion to compare the fuzzy test statistics at a specific significance level to decide whether the underlying fuzzy null hypothesis can be rejected or not. The effectiveness of the proposed two-sample test on quantiles is investigated via numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, an exponential autoregressive model for complex time series data is presented. As for estimating the parameters of this nonlinear model, a three-step procedure based on quantile methods is proposed. This quantile-based estimation technique has the benefit of being more robust compared to least/absolute squares. The performance of the introduced exponential autoregressive model is evaluated by means of four established goodness-of-fit criteria. The practical utility of the novel time series model is showcased through a comparative analysis involving simulation studies and real-world data illustrations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear time series model is developed for the case when the underlying time series data are reported by L R 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. [ABSTRACT FROM AUTHOR]

In this paper, we present a number‐between‐events (NBE) control chart for monitoring the fraction nonconforming in finite horizon production (FHP) processes and related specific performance measures. When monitoring fractions nonconforming in FHP processes, the common binomial p$p$‐chart has two crucial limitations: the underlying distributional assumptions are violated when dealing with low‐volume production and a scarce efficiency in the case of processes characterized by a low fraction nonconforming. Thus, an efficient 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 efficient 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 fixed 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 efficient monitoring of FHP processes. The proposed control chart can be conveniently used for both low‐volume and mass production in processes with frequent changeovers. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Although statistical literacy has become a key competence in today's data-driven society, it is usually not a part of statistics education. To address this issue, we propose an innovative concept for a conference-like seminar on the topic of statistical literacy. This seminar draws attention to the relevance and importance of statistical literacy, and moreover, students are made aware of the process of science communication and are introduced to the peer review process for the assessment of scientific papers. In the summer term 2020, the seminar was conducted as a joint project by the University of Hamburg, the University of Muenster, and the Joachim Herz Foundation. In this article, we present the concept of the seminar and our experience with this concept in the summer term 2020. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 (by controlling both error types). With regard to the fuzzy set theory, this test is the generalization of the conventional hypergeometric test for a proportion and provides an additional gradual consideration of the indifference zone in compliance with expert opinion or user priorities. Further, a practical application for one of the most promising resulting test types (right-tailed hypothesis testing) is carried out in statistical quality control. In particular, single sampling plans for attributes are determined via an algorithm based on the Fibonacci sequence. In addition, a parametric sensitivity analysis of calculated sampling plans is carried out in relation to changes of the lot size, consumer's and producer's risks, as well as of specified target values of quality levels. Moreover, the impact of various shapes of membership functions (piecewise linear, convex, concave, and s-shaped) is investigated by comparing respective sampling plans. [ABSTRACT FROM AUTHOR]

Purpose In 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/approach The 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.Findings The 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/value The 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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This paper gives a detailed overview of the current state of research in relation to the use of state space models and the KALMAN-filter in the field of stochastic claims reserving. Most of these state space representations are matrix-based, which complicates their applications. Therefore, to facilitate the implementation of state space models in practice, we present a scalar state space model for cumulative payments, which is an extension of the well-known chain ladder (CL) method. The presented model is distribution-free, forms a basis for determining the entire unobservable lower and upper run-off triangles and can easily be applied in practice using the KALMAN-filter for prediction, filtering and smoothing of cumulative payments. In addition, the model provides an easy way to find outliers in the data and to determine outlier effects. Finally, an empirical comparison of the scalar state space model, promising prior state space models and some popular stochastic claims reserving methods is performed. [ABSTRACT FROM AUTHOR]

Often, the claims reserves exceed the available equity of non-life insurance companies and a change in the claims reserves by a small percentage has a large impact on the annual accounts. Therefore, it is of vital importance for any non-life insurer to handle claims reserving appropriately. Although claims data are time series data, the majority of the proposed (stochastic) claims reserving methods is not based on time series models. Among the time series models, state space models combined with Kalman filter learning algorithms have proven to be very advantageous as they provide high flexibility in modeling and an accurate detection of the temporal dynamics of a system. Against this backdrop, this paper aims to provide a comprehensive review of stochastic claims reserving methods that have been developed and analyzed in the context of state space representations. For this purpose, relevant articles are collected and categorized, and the contents are explained in detail and subjected to a conceptual comparison. [ABSTRACT FROM AUTHOR]

In stochastic claims reserving, state space models have been used for almost 40 years to forecast loss reserves and to compute their mean squared error of prediction. Although state space models and the associated Kalman filter learning algorithms are very powerful and flexible tools, comparatively few articles on this topic were published during this period. Most recently, several articles have been published which highlight the benefits of state space models in stochastic claims reserving and may lead to a significant increase in its popularity for applications in actuarial practice. To further emphasize the merits of these papers, this commentary highlights various additional aspects that are useful for practical applications and offer some fruitful directions for future research. [ABSTRACT FROM AUTHOR]