A novel logarithmic framework for developing probability distributions with applications to item failures and toxic contamination datasets

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.