The idea of symmetry, which is used to describe the shape of a probability distribution, is a key concept in the theory of probability. The use of symmetric and asymmetric distributions is common in statistical inference, decision-making, and probability calculations. This article introduces a novel asymmetric model for assessing risks under a skewed claims dataset. The new distribution is also employed for both censored and uncensored validation testing. Four estimation methods, maximum likelihood, ordinary least squares, L-Moment, and Anderson Darling, were used for the risk assessment and analysis. To explain the exposure to risk within actuarial claims data, we introduced five crucial indicators, namely value-at-risk, tail-value-at-risk, tail variance, tail mean-variance, and mean excess losses. A numerical and graphical analysis is presented to assess the actuarial risk. Furthermore, the article discusses a newly developed Rao Robson Nikulin statistic for censored and uncensored validation testing. The validation testing also involved the insurance claims dataset.

Skewed probability distributions are important when modeling skewed data sets because they provide a way to describe the shape of the distribution and estimate the likelihood of extreme events. Asymmetric probability distributions have the potential to handle and assess problems in actuarial risk assessment and analysis. To that end, we present a new right-skewed one-parameter distribution. In this work and for this purpose, a right-skewed probability distribution was derived and analyzed. The new distribution outperformed the exponential distribution, the Pareto distribution, the Chen distribution, and others in the field of actuarial risk analysis. Some useful key risk indicators are considered and analyzed to analyze the risks and for comparison with the competitive model. Several actuarial risk functions and indicators are evaluated and analyzed using the U.K. insurance claims data set. The process of risk assessment and analysis was carried out using a comprehensive simulation. For the purposes of distributional validity, a modified chi-squared type test is presented and employed in the testing process. The new, modified chi-squared type test that is used is simply an extension of the Rao–Robson–Nikulin test. In this work, the distributional validity is presented and analyzed under right-skewed censored and uncensored data sets.

Classical survival models assume homogeneity among the population of individuals who are susceptible to the event of interest. However, in many practical circumstances, there is a certain amount of unobserved heterogeneity that can be caused by a variety of sources, such as environmental or genetic factors. If the heterogeneity is ignored, many issues could arise, including an overestimation of the hazard rate and inaccurate estimates of the regression coefficients. Frailty models are usually used to model the heterogeneity among individuals. In this paper, we propose a novel univariate frailty model. The frailty variable is assumed to follow the Two Parameter Lindley distribution. The maximum likelihood method is used to estimate the model parameters. The baseline hazard functions are assumed to follow Weibull, Exponential, Gompertz, and Pareto distributions, and a simulation study is performed under this assumption. We examine the characteristics of the distribution and assess its performance compared to other distributions that are frequently applied in frailty modeling by using both Nikulin-Rao-Robson and Bagdonavicius-Nikulin goodness-of-fit tests to determine the adequacy of the model. We analyze a fresh medical dataset collected from an emergency hospital in Algeria to evaluate the effectiveness and applicability of the proposed model.

We propose a new extension of the exponential distribution for right censored Bayesian and non-Bayesian distributional validation. The parameter of the new distribution is estimated using several conventional methods, including the Bayesian method. The likelihood estimates and the Bayesian estimates are compared using Pitman’s closeness criteria. The Bayesian estimators are derived using three loss functions: the extended quadratic, the Linex, and the entropy functions. Through simulated experiments, all the estimating approaches offered have been assessed. The censored maximum likelihood method and the Bayesian approach are compared using the BB algorithm. The development of the Nikulin–Rao–Robson statistic for the new model in the uncensored situation is thoroughly discussed with the aid of two applications and a simulation exercise. For the novel model under the censored condition, two applications and the derivation of the Bagdonavičius and Nikulin statistic are also described.

In this paper, we present a new exponential accelerated failure time model. Some of its properties and characterization results are derived. Different estimation methods are considered for assessing the finite sample behaviour of the estimators. Simulation studies for comparing the estimation methods are performed. Finally, we present a novel modified chi-square test for the novel exponential accelerated failure time model in both complete and right censored data cases. The validity of the new model is checked by using the theoretical global of the Nikulin-Rao-Robson. The maximum likelihood method is considered for this purpose. Two simulation studies are performed to assess the exponential accelerated failure time model and the efficiency of the Nikulin-Rao-Robson test statistic, respectively. Three real data sets are considered for illustrating the efficiency of the test statistic in validation.

We provide a new extension of the exponential distribution with an emphasis on the practical elements of the model. Six different classical estimation methods were applied and compared. The main test was evaluated on complete data using four sets of data. Additionally, four applications and the derivation of the new Nikulin statistic test for the new probability model under the censored situation are described. Both tests were evaluated through simulation experiments on complete data and on artificial and censored data. In addition, a set of simulation experiments were presented, which were used and employed to evaluate the original statistical test and the new modified statistical test based on statistical controls in the evaluation processes.