The Poisson Inverted Shifted Gompertz distribution is a new type of distribution that comes from the Poisson-generating family of distributions and is based on the "inverted shifted Gompertz distribution." This distribution has significant statistical characteristics, with clear formulations obtained mathematically. We use three distinct approaches to estimate the parameters of the distribution: maximum likelihood estimation (MLE), least-squares estimation (LSE), and Cramer-Von-Mises estimation (CVME). We undertake a simulation experiment to thoroughly assess the efficacy of maximum likelihood estimation (MLE). Empirical data from engineering and medical fields demonstrate the relevance of our suggested model. We meticulously evaluate the goodness-of-fit of the Poisson Inverted Shifted Gompertz distribution using several statistical tests and graphical techniques. Our suggested distribution demonstrates enhanced fitting capabilities and more flexibility compared to many widely used lifespan distributions. Furthermore, we investigate the Bayesian viewpoint of the proposed model by using Hamiltonian Monte Carlo (HMC) simulation methods. This detailed study gives you a full picture of how the Poisson Inverted Shifted Gompertz distribution can be used in many different areas.

Many fields of mathematics rely on convexity and nonconvexity, especially when studying optimization issues, where it stands out for a variety of practical aspects. Owing to the behavior of its definition, the idea of convexity also contributes significantly to the discussion of inequalities. The concepts of symmetry and convexity are related and we can apply this because of the close link that has grown between the two in recent years. In this study, harmonic convexity, also known as harmonic s-convexity for fuzzy number valued functions (F-NV-Fs), is defined in a more thorough manner. In this paper, we extend harmonically convex F-NV-Fs and demonstrate Hermite–Hadamard (H.H) and Hermite–Hadamard Fejér (H.H. Fejér) inequalities. The findings presented here are summaries of a variety of previously published studies.

Many authors have recently examined the relationship between symmetry and generalized convexity. Generalized convexity and symmetry have become a new area of study in the field of inequalities as a result of this close relationship. In this article, we introduce the idea of preinvex fuzzy-interval-valued functions (preinvex F∙I-V∙F) on coordinates in a rectangle drawn on a plane and show that these functions have Hermite–Hadamard-type inclusions. We also develop Hermite–Hadamard-type inclusions for the combination of two coordinated preinvex functions with interval values. The weighted Hermite–Hadamard-type inclusions for products of coordinated convex interval-valued functions discussed in a recent publication by Khan et al. in 2022 served as the inspiration for our conclusions. Our proven results expand and generalize several previous findings made in the body of literature. Additionally, we offer appropriate examples to corroborate our theoretical main findings.

Response surface methodology (RSM) refers to experimental designs for optimizing or developing processes, initially in manufacturing. In this paper, a new method is presented and an algorithm is implemented that modifies the axial part in a central composite design to achieve a good D-value and efficiency. The new designs are suitable for sequential experimentation. In comparison with known designs in the same class, the new designs are tested and found to have better D-values on a range of factors. With this new approach, efficient orthogonal designs for response surface methodology were generated for a number of parameters that were previously impossible to construct. The new generated designs and their comparison with known designs from the literature are presented in tables for practitioners’ use.

Peaks over a random threshold value-at-risk (PORT-VaR) analysis is a powerful tool for evaluating extreme value reliability data, particularly for materials like carbon and glass fibers. By incorporating random thresholds into traditional value at risk (VaR) and tail value at risk (TVaR) frameworks, it provides a more nuanced understanding of how materials behave under extreme conditions, making it invaluable for applications where failure is costly or dangerous, such as aerospace, automotive, and civil engineering. The combination of Mean of Order P (MO-P), VaR and PORT-VaR analyses in medical data offers important insights into risk evaluation and patient management. By examining both average and extreme strength of glass fibers, healthcare professionals can create more effective treatment plans, enhance patient outcomes, and improve overall care quality. This comprehensive approach enables more sophisticated decision-making and targeted interventions in clinical settings. To illustrate our main objective and conduct a medical analysis, we introduced a new extreme value model called the generalized Rayleigh reciprocal-Weibull (GR-RW) and presented its key mathematical results. Additionally, we conducted a simulation study and analyzed two real datasets to compare the competing models.

In this piece of work, we examine and present a completely new discrete family of distributions that we have created. Our investigation into the relevant mathematical properties and characterizations of the system makes use of both analytical and numerical methods. We focus on a particular member of this family so that we can study its theoretical foundations as well as its graphical and numerical representations. This new model contains a few different hazard rate functions, some of which are referred to as "increasing constant", "decreasing-constant-increasing (U)", "constant", "U-constant", "decreasing", and "J-shape" In a similar vein, the model's probability mass function provides a variety of forms, all of which are helpful and practical. These forms include "asymmetric left skewed," "right skewed with wide peak," "right skewed," "bimodal," "symmetric," and "right skewed," amongst others. Each of these forms is valuable and applicable in their own way. These forms might be discovered in the probability mass function that the model generates. In this investigation, in addition to the Bayesian estimating technique under the traditional loss function of squared errors, we investigate and make use of a total of eight estimate strategies that are not founded on Bayesian theory (classical methods). Simulations employing the Markov Chain Monte-Carlo method are run for comparing the Bayesian way of estimation with the more traditional approach of estimating values. According to the findings that we've compiled, the estimation strategy that is referred to as maximum likelihood yields the most accurate results across the board and for all different types of sample sizes. In addition, we evaluate and contrast the various methods of estimation by making use of six distinct real dataset sets; this indicates the versatility of the unique model that we have developed.