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The Weibull distribution is an important continuous distribution that is cardinal in reliability analysis and lifetime modeling. On the other hand, it has several limitations for practical applications, such as modeling lifetime scenarios with non-monotonic failure rates. However, accurate modeling of non-monotonic failure rates is essential for achieving more accurate predictions, better risk management, and informed decision-making in various domains where reliability and longevity are critical factors. For this reason, we introduce a new three parameter lifetime distribution—the Modified Kies Weibull distribution (MKWD)—that is able to model lifetime scenarios with non-monotonic failure rates. We analyze the statistical features of the MKWD, such as the quantile function, median, moments, mean, variance, skewness, kurtosis, coefficient of variation, index of dispersion, moment generating function, incomplete moments, conditional moments, Bonferroni, Lorenz, and Zenga curves, and order statistics. Various measures of uncertainty for the MKWD such as Rényi entropy, exponential entropy, Havrda and Charvat entropy, Arimoto entropy, Tsallis entropy, extropy, weighted extropy and residual extropy are computed. We discuss eight different parameter estimation methods and conduct a Monte Carlo simulation study to evaluate the performance of these different estimators. The simulation results show that the maximum likelihood method leads to the best results. The effectiveness of the newly suggested model is demonstrated through the examination of two different sets of real data. Regression analysis utilizing survival times data demonstrates that the MKWD model offers a superior match compared to other current distributions and regression models. [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 recent emergence and widespread adoption of Artificial Intelligence (AI) across various industries have not only increased efficiency and innovation but also introduced complex security challenges. Addressing these security challenges is an ever‐evolving area of research that seeks to mitigate issues affecting a wide range of individuals and sectors. Understanding the threats and vulnerabilities posed by these systems, and how to effectively defend against them, has become more crucial than ever. This paper aims to provide an overview of the most common attack vectors and their respective defense strategies, focusing particularly on those relevant to Industry 4.0. A major area of interest is adversarial machine learning, a relatively new field that focuses on corrupting, confusing, and manipulating AI models by intervening in different phases of their life cycle. The key findings indicate that FLAME offers the best protection against data poisoning attacks for neural network image detection models in a federated learning environment. Additionally, due to the specific threat model in Industry 4.0, defensive distillation emerges as the most promising defense strategy against evasion attacks. [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]

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 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]

In machine learning, the fused lasso is a regularization technique that is used to handle problems where the underlying signal has some kind of structure. In this paper, we extend fused lasso estimation for regression models characterized by fuzzy predictors and fuzzy responses. We present the first fused lasso regression model that is able to handle and analyze high-dimensional fuzzy data. The proposed model provides feature selection, leads to an improved predictive performance, and ensures interpretable and explainable models, all while being computationally efficient. Moreover, it can recover the true signal more accurately, find solutions that are structurally sparse, and is robust to noise. We conduct comprehensive comparative analysis and demonstrate the practical applicability of the presented fuzzy regression model through simulation and real-life applications. [ABSTRACT FROM AUTHOR]

When dealing with nonparametric regression problems, kernel-based methods are often employed. In this paper, we investigate nonparametric regression problems with fuzzy responses and exact predictors by implementing an extended center and range method within a three-stage procedure. For this purpose, we utilize the Nadaraya–Watson estimator for estimating the unknown fuzzy smooth function. In each stage, the unknown bandwidth is determined with the help of a generalized cross validation criterion. To evaluate and to compare the performance of our proposed fuzzy nonparametric regression model with established regression models, different goodness-of-fit criteria are considered. Finally, we investigate the practical applicability and show the superiority of the proposed regression model by means of a simulation study and real data applications. • We develop a nonparametric regression model for fuzzy responses and exact predictors. • The nonlinear fuzzy smooth function is estimated within a three-stage procedure. • In each stage, the unknown bandwidth is evaluated by a generalized cross validation criterion. • We investigate different kernel functions and examine the model performance by various goodness-of-fit criteria. • We perform comparative analysis through simulation and real-life applications. [ABSTRACT FROM AUTHOR]