Numerous adaptations of traditional entropy concepts and their residual counterparts have emerged in statistical research. While some methodologies incorporate supplementary variables or reshape foundational assumptions, many ultimately align with conventional formulations. This study introduces a novel extension termed residual cumulative generalized exponential entropy to broaden the scope of residual cumulative entropy for continuous distributions. Key attributes of the proposed measure include non-negativity, bounds, its relationship to the continuous entropy measure, and stochastic comparisons. Practical implementations are demonstrated through case studies involving established probability models. Additionally, insights into order statistics are derived to characterize the measure's theoretical underpinnings. The residual cumulative generalized exponential entropy framework bridges concepts such as Bayesian risk assessment and excess wealth ordering. For empirical implementation, non-parametric estimation strategies are devised using data-driven approximations of residual cumulative generalized exponential entropy, with two distinct estimators of the cumulative distribution function evaluated. A practical application is showcased, using clinical diabetes data. The study further explores the role of generalized exponential entropy in identifying distributional symmetry, mainly through its application to uniform distributions to pinpoint symmetry thresholds in ordered data. Finally, the utility of generalized exponential entropy is examined in pattern analysis, with a diabetes dataset serving as a benchmark for evaluating its classification performance.

Nowadays, obesity is recognized as a worldwide epidemic that has become a major cause of death and comorbidities. Recommending appropriate treatment is critical in the global health environment. For obesity treatment to be effective, the person must be able to follow a specific diet that meets his needs so that he can follow it for a long time or forever to maintain fitness. This research aims to determine the best diet among the trusted diets for every person based on his needs and circumstances. This occurs when applying a decision-making technique based on the effective fuzzy soft multiset concept. For this purpose, the definition of the effective fuzzy soft multiset as well as its types, operations, and properties are introduced. Furthermore, a decision-making method is proposed based on the effective fuzzy soft multiset environment. Using matrices operations, one can easily apply the decision-making process based on this new extension of sets to choose the optimal diet for everyone. Finally, an extensive comparative analysis of the previous methods is undertaken and also summarized in a chart to attract focus on the benefits of the suggested algorithm and to demonstrate how they differ from the current one.

The fractional generalized cumulative residual entropy, a broader version of the cumulative residual entropy, holds significance in assessing the uncertainty model of random variables and maintains straightforward connections with reliability models and crucial information. This article represents and modifies some novel features of the fractional generalized cumulative residual entropy and discusses the weak convergence. Additionally, the measure is utilized to assess uniformity, involving the derivation of the limit distribution and an approximation of the test statistic's distribution. Furthermore, the concept of stability is addressed. Moreover, the presentation includes the critical points and power analysis against alternative distributions of this test statistic. Furthermore, a simulation study is carried out to compare the power value of the proposed test with that of other tests of uniformity. Moreover, the uniformity test utilizes real data on daily smokers in the countries of the Euro Area. Finally, our model's exponential distribution is applied to our model's empirical form.