This study introduces a new extension of the unit Weibull distribution called the unit power generalized Weibull distribution (UPGWD). The UPGWD arises from inverse exponential function transformation of the power generalized Weibull distribution. It is a highly competitive distribution compared with the existing unit distributions in the literature, offering significant flexibility. The probability density function of the UPGWD can display several forms, including constant, bathtub, unimodal, J-shaped (increasing), and inverted J-shaped (decreasing) configurations. Conversely, its hazard function may exhibit increasing J-shaped and bathtub configurations. Some of its corresponding basic statistical and reliability properties are introduced. Furthermore, the maximum likelihood estimation (MLE) technique is applied to estimate its parameters. A Monte Carlo simulation study is performed to assess the accuracy of the MLE estimates. Finally, to demonstrate the potential importance of the UPGWD, four applications with actual lifetime data related to COVID-19, reliability, engineering, and radiation are discussed. The empirical application further validated its efficacy, surpassing the earlier existing unit Weibull distributions, including the unit Weibull, the unit inverted exponentiated Weibull, the upper truncated Weibull, the bounded exponentiated Weibull, the power upper truncated Weibull, and the Poisson unit Weibull distributions.

The modeling of bivariate data in statistics often requires constructing families of bivariate distributions with predefined marginals. In this study, we introduced a novel bivariate distribution, denoted as EP-WD-SAR, which combines the Sarmanov (SAR) copula with the Epanechnikov-Weibull marginal distribution (EP-WD). We analyzed its statistical properties, including product moments, correlation coefficient, moment-generating function, conditional distribution, and concomitants of order statistics. Additionally, we evaluated key reliability and information measures such as the hazard function, reversed hazard function, bivariate extropy, bivariate weighted extropy, and bivariate cumulative residual extropy. Parameter estimation was performed using maximum likelihood, asymptotic confidence intervals, and Bayesian methods. Finally, we demonstrated the advantages of the EP-WD-SAR model over existing alternatives, including the bivariate Weibull-SAR, bivariate Epanechnikov-exponential-SAR, bivariate exponential-SAR, and bivariate Chen-SAR distributions through applications to real data sets.

In statistical modeling, bivariate models are essential, especially when examining data with two associated variables. Bivariate distributions capture dependencies between variables, offering a more realistic depiction of real-world phenomena compared to univariate models that treat variables independently. This is particularly important in domains where variables frequently show non-trivial correlations, such as environmental science, reliability engineering, medicine, and finance. This motivates the proposal of a bivariate distribution that uses the generalized Farlie-Gumbel-Morgenstern (FGM) copula and Weibull marginal distribution, referred to as the GFGM-WD. The GFGM-WD describes bivariate lifetime data with weak to moderate correlation between variables. The suggested model was employed to investigate the reliability of dependent stress-strength models. Several properties of the GFGM-WD were derived, including the product moment, the coefficient of correlation between the inner variables, and the conditional expectation. Additionally, the statistical characteristics of the concomitants' k-record values from the GFGM-WD were discussed. We ran comprehensive Monte Carlo simulations to assess the suggested distribution's performance and used the maximum likelihood estimation and Bayesian methods to estimate its parameters. Finally, the distribution was tested on two actual medical datasets, showing that it outperformed other pre-existing bivariate models in terms of fitting accuracy.