Adaptive Lifetimes with Properties and Applications: Unleashing Flexibility in Survival and Reliability Models

Most processes in the world display a variety of uncertain random behaviors, and to describe this unpredictable randomness, one needs a robust probability model that can grasp the underlying behavior of the data with the utmost competence and in the most organized manner. Therefore, for practitioners, the suggestion or adjustment of lifetime distributions is of the utmost importance. In this paper, we have developed a novel lifetime distribution by compounding a new generalized two-parameter Lindley distribution with power series distributions. The proposed family is versatile encompassing notable lifetime sub-models such as the Generalized two-parameter Lindley–Poisson (GTPLP), Generalized two-parameter Lindley Geometric (GTPLG), Generalized two-parameter Lindley Logarithmic (GTPLL), and Generalised two-parameter Lindley Binomial (GTPLB) distributions, while also offering a broad range of applicability. The necessary structural properties like moment generating function, order statistics, and, most importantly, quantile and mean residual task computations involving the Lambert W function have also been performed. The discussion also includes parameter estimation using maximum likelihood estimation and the ordinary least squares technique. We exclusively calculate Galton skewness and Moors kurtosis for the GTPLL distribution. We simulated GTPLL distribution and explored applications of the proposed family on three-lifetime data sets. The statistical analysis delicately exposes the beauty and broad flexibility of the proposed family of lifetime distributions. In addition, we discussed GTPLPS to solve the reliability optimization problem for communication systems using the Lagrange multiplier technique. Improved system performance is evidence that reliability optimization is effective. Regarding failure rates, the GTPLP distribution is more reliable for modelling and analyzing systems as compared to GTPLG and GTPLL.