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The extropy has recently been introduced as the dual concept of entropy. Moreover, in the context of the Dempster–Shafer evidence theory, Deng studied a new measure of discrimination, named the Deng entropy. In this paper, we define the Deng extropy and study its relation with Deng entropy, and examples are proposed in order to compare them. The behaviour of Deng extropy is studied under changes of focal elements. A characterization result is given for the maximum Deng extropy and, finally, a numerical example in pattern recognition is discussed in order to highlight the relevance of the new measure.

Deng entropy and extropy are two measures useful in the Dempster–Shafer evidence theory (DST) to study uncertainty, following the idea that extropy is the dual concept of entropy. In this paper, we present their fractional versions named fractional Deng entropy and extropy and compare them to other measures in the framework of DST. Here, we study the maximum for both of them and give several examples. Finally, we analyze a problem of classification in pattern recognition in order to highlight the importance of these new measures.

This paper introduces and studies a new generalization of cumulative past extropy called weighted cumulative past extropy (WCPJ) for continuous random variables. We explore the following: if the WCPJs of the last order statistic are equal for two distributions, then these two distributions will be equal. We examine some properties of the WCPJ, and a number of inequalities involving bounds for WCPJ are obtained. Studies related to reliability theory are discussed. Finally, the empirical version of the WCPJ is considered, and a test statistic is proposed. The critical cutoff points of the test statistic are computed numerically. Then, the power of this test is compared to a number of alternative approaches. In some situations, its power is superior to the rest, and in some other settings, it is somewhat weaker than the others. The simulation study shows that the use of this test statistic can be satisfactory with due attention to its simple form and the rich information content behind it.

The purpose of the paper is to introduce the Jensen–inaccuracy measure and examine its properties. Furthermore, some results on the connections between the inaccuracy and Jensen–inaccuracy measures and some other well-known information measures are provided. Moreover, in three different optimization problems, the arithmetic mixture distribution provides optimal information based on the inaccuracy information measure. Finally, two real examples from image processing are studied and some numerical results in terms of the inaccuracy and Jensen–inaccuracy information measures are obtained.

In the literature, estimation of weighted extropy is infrequently considered. In this paper, some non-parametric estimators of weighted extropy are given. The validation and comparison of the estimators are implemented with the help of simulation study and data illustration. The usefulness of the estimators is demonstrated using real data sets.

Tsallis introduced a non-logarithmic generalization of Shannon entropy, namely Tsallis entropy, which is non-extensive. Sati and Gupta proposed cumulative residual information based on this non-extensive entropy measure, namely cumulative residual Tsallis entropy (CRTE), and its dynamic version, namely dynamic cumulative residual Tsallis entropy (DCRTE). In the present paper, we propose non-parametric kernel type estimators for CRTE and DCRTE where the considered observations exhibit an ρ-mixing dependence condition. Asymptotic properties of the estimators were established under suitable regularity conditions. A numerical evaluation of the proposed estimator is exhibited and a Monte Carlo simulation study was carried out.

Wire Arc Additive Manufacturing (WAAM) represents a disruptive technology in the field of metal additive manufacturing. Understanding the relationship between input factors and layer geometry is crucial for studying the process comprehensively and developing various industrial applications such as slicing software and feedforward controllers. Statistical tools such as clustering and multivariate polynomial regression provide methods for exploring the influence of input factors on the final product. These tools facilitate application development by helping to establish interpretable models that engineers can use to grasp the underlying physical phenomena without resorting to complex physical models. In this study, an experimental campaign was conducted to print steel components using WAAM technology. Advanced statistical methods were employed for mathematical modeling of the process. The results obtained using linear regression, polynomial regression, and a neural network optimized using the Tree-structured Parzen Estimator (TPE) were compared. To enhance performance while maintaining the interpretability of regression models, clusterwise regression was introduced as an alternative modeling technique along with multivariate polynomial regression. The results showed that the proposed approach achieved results comparable to neural network modeling, with a Mean Absolute Error (MAE) of 0.25 mm for layer height and 0.68 mm for layer width compared to 0.23 mm and 0.69 mm with the neural network. Notably, this approach preserves the interpretability of the models; a further discussion on this topic is presented as well.

Necrotizing fasciitis (NF) is a rare but aggressive soft tissue infection with high rates of mortality and amputation, making early identification of key prognostic biomarkers essential for clinical management. However, the rarity and heterogeneity of NF mean clinical datasets are often small and non-normally distributed, limiting the effectiveness of standard parametric statistical approaches. To address this, we retrospectively analyzed 66 NF patients using a robust, distribution-free framework that combines the Nonparametric Combination (NPC) methodology and bootstrap resampling. We specifically assessed glycated hemoglobin (HBA1C) and serum albumin (ALBUMINA) as potential predictors of two outcomes: mortality (MORTO) and major amputation (AMPUTAZIONE). NPC enabled exact multivariate hypothesis testing while rigorously controlling the family-wise error rate (FWER), and bootstrap resampling generated 95% confidence intervals (CI) for critical biomarkers. HBA1C was an exceptionally significant predictor compared to the 7.0% clinical threshold (p = 1.04 × 10−154, CI: 0.0830–0.0957), while ALBUMINA showed greater biological variability but no significant association with outcomes (2.8 g/dL; p = 0.267, CI: 2.551–2.866). We also developed a global severity ranking, integrating multiple variables to improve clinical risk stratification. Our results demonstrate that permutation-based and resampling methods provide reliable, actionable insights from challenging small-sample clinical datasets. Based on a small-sample dataset from necrotizing fasciitis patients, this framework provides a replicable model for robust, nonparametric statistical analysis in similarly rare and high-risk medical conditions. This study introduces a Nonparametric Combination (NPC) framework for risk scoring in necrotizing fasciitis using bootstrap resampling and permutation tests. Key predictors like HBA1C and Albumin were assessed, achieving an AUC of 0.89 and a Youden Index of 0.71. The model offers a robust, interpretable tool for clinical risk stratification in small-sample rare disease settings.

In scientific analyses, measurement errors in data can significantly impact statistical inferences, and ignoring them may lead to biased and invalid results. This study focuses on the estimation of the residual extropy function, in the presence of measurement errors. We developed an estimator for the extropy function and established its asymptotic properties. A comprehensive simulation study evaluates the performance of the proposed estimators under various error scenarios, while their practical utility and precision are demonstrated through an application to a real-world data set.

We consider dynamic versions of the mutual information of lifetime distributions, with a focus on past lifetimes, residual lifetimes, and mixed lifetimes evaluated at different instants. This allows us to study multicomponent systems, by measuring the dependence in conditional lifetimes of two components having possibly different ages. We provide some bounds, and investigate the mutual information of residual lifetimes within the time-transformed exponential model (under both the assumptions of unbounded and truncated lifetimes). Moreover, with reference to the order statistics of a random sample, we evaluate explicitly the mutual information between the minimum and the maximum, conditional on inspection at different times, and show that it is distribution-free in a special case. Finally, we develop a copula-based approach aiming to express the dynamic mutual information for past and residual bivariate lifetimes in an alternative way.