Several higher education institutions recommended active learning many decades ago. In many fields, artificial intelligence (AI) has turned out to be a dominant aspect of people's lives. One of the fields that welcome AI is education. AI can play a crucial role in active learning, which is a teaching method that encourages students to take an active role in their learning process. Active learning can involve a wide range of activities, such as problem-solving, group discussions, and self-reflection. This quantitative research paper aims to explore students' engagement and perceptions of patterns of embracing AI tools and their relationship to student learning outcomes. A diverse sample of 355 students from a highly reputable university in the United Arab Emirates participated in the research study (UAE). Descriptive Analysis and Correlation Coefficient were used to achieve the paper's objectives. Findings uncovered students’ high engagement level in classroom activities, and they perceived AI tools as useful and effortless, which promotes their engagement and attention inside classrooms. Furthermore, the results demonstrated the existence of a positive association between students' perceptions of embracing AI and student engagement in the learning environment. However, students’ learning outcomes have a non-remarkable association with student engagement and their perception of AI patterns. The study's findings suggest embracing and promoting AI applications in classrooms to keep students engaged. This study's use of Neutrosophic sets offers a fresh way to approach the problem of ambiguity when evaluating students' opinions and performance in relation to AI technologies. Neutrosophic sets, a mathematical framework designed to manage uncertainty, provide a sophisticated way to understand how students' complicated interactions with AI operate. This integration promises a more comprehensive and flexible educational paradigm and is a trailblazing method of negotiating the complexities of active learning in higher education.

This paper explores the properties of a family of bivariate copulas based on a new approach using the counter-monotonic shock method. The resulting copula covers the full range of negative dependence induced by one parameter. Expressions for the copula and density are derived and many theoretical properties are examined thoroughly, including explicit expressions for prominent measures of dependence, namely Spearman’s rho, Kendall’s tau and Blomqvist’s beta. The convexity properties of this copula are presented, together with explicit expressions of the mixed moments. Estimation of the dependence parameter using the method of moments is considered, then a simulation study is carried out to evaluate the performance of the suggested estimator. Finally, an application of the proposed copula is illustrated by means of a real data set on air quality in New York City.

We introduce a new family of bivariate exponential distributions based on the counter-monotonic shock model. This family of distribution is easy to simulate and includes the Fréchet lower bound, which allows to span all degrees of negative dependence. The construction and distributional properties of the proposed bivariate distribution are presented along with an estimation of the parameters involved in our model based on the method of moments. A simulation study is carried out to evaluate the performance of the suggested estimators. An extension to the general model describing both negative and positive dependence is sketched in the last section of the paper.

Investigating dependence structures across various fields holds paramount importance. Consequently, the creation of new copula families plays a crucial role in developing more flexible stochastic models that address the limitations of traditional and sometimes impractical assumptions. The present article derives some reasonable conditions for validating a copula of the ratio-type form $uv/(1-\theta f(u\left)g\right(v\left)\right)$. It includes numerous examples and discusses the admissible range of parameter $\theta$, showcasing the diversity of copulas generated through this framework, such as Archimedean, non-Archimedean, positive dependent, and negative dependent copulas. The exploration extends to the upper bound of a general family of copulas, $uv/(1-\theta \varphi (u,v\left)\right)$, and important properties of the copula are discussed, including singularity, measures of association, tail dependence, and monotonicity. Furthermore, an extensive simulation study is presented, comparing the performance of three different estimators based on maximum likelihood, $\rho$-inversion, and the moment copula method.

Variables that have revived new interest through computational developments and extensive data analysis. This article contributes to the subject by generalizing the bivariate copula introduced recently in8 and based on the concept of the counter-monotonic shock method. The proposed copula has the feature of covering the full range of negative dependence induced by two dependence parameters, which is not so common in the specialized literature. We examine the main characteristics of this copula. In particular, the absolutely continuous and singular copula components are derived. Analytical expressions of important concordance measures, such as Spearman’s rho and Kendall’s tau, are established, along with expressions of the product moments. A real neutrosophic data set, based on the daily quality of air in the New York Metropolitan Area, is used to illustrate the applicability of the proposed copula, with quite convincing results.

In statistical literature, several dependence measures have been extensively established and treated, including Pearson's correlation coefficient, Spearman's ρ and Kendall's τ. In the context of survival analysis with length-biased data, a measure of dependence between survival time and covariates appears to have not received much intention in the literature. The purpose of this paper is to extend Kent's [Information gain and a general measure of correlation. Biometrika. 1983;70(1):163–173.] dependence measure, based on the concept of information gain, to length-biased survival data. Specifically, we develop a new approach to measure the degree of dependence between survival time and several continuous covariates, without censoring, when the relationship is linear. In this regard, kernel density estimation with a regression procedure is proposed. The consistency for all proposed estimators is established. In particular, the performance of the dependence measure for length-biased data is investigated by means of simulations studies.