After defining a new log-logistic model and studying its properties, some new bivariate type versions using “Farlie-Gumbel-Morgenstern Copula”, “modified Farlie-Gumbel-Morgenstern Copula”, “Clayton Copula”, and “Renyi’s entropy Copula” are derived. Then, using the Bagdonavicius-Nikulin goodness-of-fit (BN-GOF) test for validation, we proposed a goodness-of-fit test for a new log-logistic model. The modified test is applied for the “right censored” real dataset of survival times. All elements of the modified test are explicitly derived and given. Three real data applications are presented for measuring the flexibility and the importance of the new model under the uncensored scheme. Two other real datasets are analyzed for censored validation.

Analyzing the future values of anticipated claims is essential in order for insurance companies to avoid major losses caused by prospective future claims. This study proposes a novel three-parameter compound Lomax extension. The new density can be "monotonically declining", "symmetric", "bimodal-asymmetric", "asymmetric with right tail", "asymmetric with wide peak" or "asymmetric with left tail". The new hazard rate can take the following shapes: "J-shape", "bathtub (U-shape)", "upside down-increasing", "decreasing-constant", and "upside down-increasing". We use some common copulas, including the Farlie-Gumbel-Morgenstern copula, the Clayton copula, the modified Farlie-Gumbel-Morgenstern copula, Renyi's copula and Ali-Mikhail-Haq copula to present some new bivariate quasi-Poisson generalized Weibull Lomax distributions for the bivariate mathematical modelling. Relevant mathematical properties are determined, including mean waiting time, mean deviation, raw and incomplete moments, residual life moments, and moments of the reversed residual life. Two actual data sets are examined to demonstrate the unique Lomax extension's usefulness. The new model provides the lowest statistic testing based on two real data sets. The risk exposure under insurance claims data is characterized using five important risk indicators: value-at-risk, tail variance, tail-value-at-risk, tail mean-variance, and mean excess loss function. For the new model, these risk indicators are calculated. In accordance with five separate risk indicators, the insurance claims data are employed in risk analysis. We choose to focus on examining these data under five primary risk indicators since they have a straightforward tail to the left and only one peak. All risk indicators under the insurance claims data are addressed for numerical and graphical risk assessment and analysis.

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.

This study introduces and examines a new probability distribution, presenting various characterizations. Key financial risk measures, including the value-at-risk (VaR), tail-value-at-risk (TVaR), also referred to as conditional tail expectation or conditional value-at-risk (CVaR), tail variance (TV), tail mean-variance (TMV), and mean excess loss (MExL) function are evaluated using maximum likelihood estimation, ordinary least squares, weighted least squares, and the Anderson-Darling estimation methods. These methods are applied for actuarial analysis in both a simulation study and an insurance claims data application. For validation of the distribution using complete data, the widely recognized Nikulin-Rao-Robson statistic is utilized and assessed through simulations and three real data sets. Two uncensored real-life data sets for failure times and remission times are used in uncensored validation. Additionally, for censored data validation, a modified version of the Nikulin-Rao-Robson statistic is proposed and evaluated through extensive simulations and three censored real data sets.