We introduce a new flexible statistical model for disability and reliability case studies, designed to enhance modeling capabilities in reliability engineering and disability prevalence analysis. This model extends the classical Pareto Type Ⅱ distribution by incorporating additional shape parameters, allowing for greater flexibility in modeling various hazard rate shapes and tail behaviors, particularly light tails. Essential properties of the model are explored. The model's applicability and risk assessment potential are demonstrated through case studies: one focusing on reliability engineering (aircraft windshield failure and service times) and another analyzing disability prevalence rates in Saudi Arabia for 2016. Advanced tail analysis tools such as the Hill estimator, Value-at-Risk (VaR), and Peaks Over a Random Threshold (PORT) were employed. The analyses reveal that the proposed OGPTII model offers superior goodness-of-fit compared to several existing models for both reliability and Saudi Arabia disability data, accurately captures the light-tailed nature of the datasets, and provides crucial insights into potential risk factors. This makes it a valuable tool for predictive maintenance strategies in engineering and for evidence-based policy planning and resource allocation in the context of disability management.

This study introduces the Inverse Burr-X Burr-XII (IBXBXII) distribution as a novel approach for handling asymmetric-bimodal claims and revenues. It explores the distribution's statistical properties and evaluates its performance in three contexts. The analysis includes assessing entropy, highlighting the distribution's significance in various fields, and comparing it to rival distributions using practical examples. The IBXBXII model is then applied to analyze risk indicators in actuarial data, focusing on bimodal insurance claims and income. Simulation analysis shows its preference for right-skewed data, making it suitable for mathematical modeling and actuarial risk assessments. The study emphasizes the IBXBXII model's versatility and effectiveness, suggesting it as a flexible framework for actuarial data analysis, particularly in cases of large samples and right-skewed data.

It is often desired for practical reasons to cease a life test at a predetermined period ${𝓽}_0$. In this study, we provide sequential inspection sampling plan (SISP) for amputated life tests underlying the Burr Ⅻ (BTXII) distribution. We considered the ${\rho }^{\mathrm{t}\mathrm{h}}$ percentile lifetime of a product as the quality parameter. The sequential sampling plan is a dynamic and efficient technique to quality control and statistical decision-making. Unlike fixed sample plans, which require a predetermined number of samples before deciding, sequential plan enables ongoing analysis as a batch is examined. This flexible plan enables decisions- such as rejecting, accepting or continuing the sampling process - to be made after each sample, which is cumulative data for the number of nonconforming items. Acceptance, rejection limit lines, and optimal sample size at levels of manufacturer and customer errors were analyzed. A technique is provided for calculating the operation characteristic (OC) function and average sample number (ASN) in the suggested SISP. The effectiveness of the SISP was compared to the single, double, and repeating sampling strategies. In comparison to the single, double, and repetitive acceptance sampling plans (RASPs) the proposed sequential sampling acceptance strategy requires fewer sample resources on average for amputated life testing. The suggested method's uses are demonstrated with illustrated instances. In industrial applications, two actual sets of data are employed to demonstrate the SISP's flexibility.