A new family of distributions, the Log-generator class, is introduced in this work, deriving its basis from the logarithmic function. The study establishes formal expressions for the generator's probability density function. Subsequently, the New Log-Rayleigh distribution (NLRD) is formulated, selecting the Rayleigh distribution as its foundation and concentrating on a particular case within the proposed family. Several distributional properties, including moments, reliability indices, entropies, and order statistics, are derived through systematic mathematical approaches. The finite-sample behavior and effectiveness of the parameters are assessed through simulation experiments, analyzing the bias, mean square error, and mean relative error. To validate its practical utility and highly adaptive right-skewed tail flexibility, the proposed distribution is applied to real-world datasets concerning repairable system failures and groundwater contamination from vinyl chloride, demonstrating its strong capability to model highly skewed empirical profiles. Furthermore, a group acceptance sampling strategy (GASP) for quality assurance is applied to the formulated model.

This study introduces the type-I heavy-tailed Burr XII (TIHTBXII) distribution, a highly flexible and robust statistical model designed to address the limitations of conventional distributions in analyzing data characterized by skewness, heavy tails, and diverse hazard behaviors. We meticulously develop the TIHTBXII’s mathematical foundations, including its probability density function (PDF), cumulative distribution function (CDF), and essential statistical properties, crucial for theoretical understanding and practical application. A comprehensive Monte Carlo simulation evaluates four parameter estimation methods: maximum likelihood (MLE), maximum product spacing (MPS), least squares (LS), and weighted least squares (WLS). The simulation results consistently show that as sample sizes increase, the Bias and RMSE of all estimators decrease, with WLS and LS often demonstrating superior and more stable performance. Beyond theoretical development, we present a practical application of the TIHTBXII distribution in constructing a group acceptance sampling plan (GASP) for truncated life tests. This application highlights how the TIHTBXII model can optimize quality control decisions by minimizing the average sample number (ASN) while effectively managing consumer and producer risks. Empirical validation using real-world datasets, including “Active Repair Duration,” “Groundwater Contaminant Measurements,” and “Dominica COVID-19 Mortality,” further demonstrates the TIHTBXII’s superior fit compared to existing models. Our findings confirm the TIHTBXII distribution as a powerful and reliable alternative for accurately modeling complex data in fields such as reliability engineering and quality assessment, leading to more informed and robust decision-making.