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.

This paper investigates a novel category of probability distributions and a specific member within this category. We have formulated a new family of trigonometric distributions by utilizing the odds ratio derived from the distribution function of a base distribution. This newly devised distribution family termed the “Sine pie-power odd-G family” of distributions, is constructed through a transformation involving the sine function. The paper presents an overview of the fundamental characteristics inherent to this proposed distribution family. Using the Weibull distribution as a base reference, we have introduced a member belonging to the proposed distribution family. This member demonstrates various hazard functions such as j, reverse-j, increasing, decreasing, or bathtub shapes. The paper examines essential statistical attributes of this newly introduced distribution. The estimation of the distribution’s parameters is carried out via the maximum likelihood estimation method. The accuracy of the parameter estimation procedure is validated through Monte Carlo simulations. The outcomes of these simulations reveal a reduction in biases and mean square errors as sample sizes increase, even for small samples. Two sets of real-engineering data are considered to demonstrate the proposed distribution’s applicability. The performance of the suggested distribution is evaluated using some model selection criteria and goodness-of-fit test statistics. Empirical evidence from these evaluations substantiates that the proposed model outperforms six existing models.

Frailty models are important for survival data because they allow for the possibility of unobserved heterogeneity problem. The problem of heterogeneity can be existed due to a variety of factors, such as genetic predisposition, environmental factors, or lifestyle choices. Frailty models can help to identify these factors and to better understand their impact on survival. In this study, we suggest a novel quasi xgamma frailty (QXg-F) model for the survival analysis. In this work, the test of Rao–Robson and Nikulin is employed to test the validity and suitability of the probabilistic model, we examine the distribution’s properties and evaluate its performance in comparison with many relevant cox-frailty models. To show how well the QXg-F model captures heterogeneity and enhances model fit, we use simulation studies and real data applications, including a fresh dataset gathered from an emergency hospital in Algeria. According to our research, the QXg-F model is a viable replacement for the current frailty modeling distributions and has the potential to improve the precision of survival analyses in a number of different sectors, including emergency care. Moreover, testing the ability and the importance of the new QXg-F model in insurance is investigated using simulations via different methods and application to insurance data.

In recent years, integrating trigonometric techniques into probability models has garnered significant interest. This paper presents a novel trigonometric generator based on the Arc cosine function, referred to as the Arc cos-\(\Psi\) distribution. The proposed distribution demonstrates unique and flexible patterns in its probability density function (PDF) and hazard rate function (HRF), showcasing its ability to effectively model both symmetrical and asymmetrical data behaviors. Key mathematical properties of the distribution are thoroughly investigated, including moments, extremum behavior of the PDF and HRF, incomplete moments, quantile function, and entropies. Parameter estimation is carried out using various methods, and their performance is assessed through comprehensive numerical studies. Additionally, a simulation study is conducted to further validate the distribution’s properties and estimation techniques. The practical utility and adaptability of the model are demonstrated using two real-world datasets, including COVID-19 data, where the distribution provides an exceptional fit and reveals unique data characteristics. This underscores its potential for modeling complex datasets with intricate structures, making it a valuable addition to the statistical toolkit.

This research introduces a novel two-parameter distribution, the power-modified XLindley distribution, developed through the application of power transformation techniques to the existing modified XLindley distribution. This new distribution enhances flexibility and adaptability in statistical modeling. We conduct a thorough examination of its statistical properties, exploring its potential to improve data fitting and modeling accuracy. To assess the effectiveness of the model, we employ multiple estimation techniques and evaluate their performance through extensive simulation experiments. Our findings indicate that the maximum product of the spacings method is particularly effective for parameter estimation. To demonstrate the practical utility of the proposed model, we apply it to two real-world datasets: one related to flood data and the other to reliability engineering. The results underscore the distribution’s superior ability to capture the characteristics of these datasets compared to existing models, highlighting its significance for applications in natural disaster analysis and reliability studies.