Two-Sided Length-Biased Wald Distribution: Properties, Estimation, and Applications

This article introduces the two-sided length-biased Wald (TSLBW) distribution, an extension of the length-biased Wald (LBW) distribution, designed to model processes in which failure or event completion can originate from both sides. The TSLBW distribution is particularly suited to reliability engineering, survival analysis, and material failure studies where crack propagation or failure initiation may occur simultaneously from two directions. Notably, the distribution is right-skewed and non-symmetric, yet it arises naturally from the symmetric propagation of cracks from both sides, bridging distributional asymmetry with the underlying symmetry of the failure initiation process. This interplay between symmetry and asymmetry motivates its study within the broader framework of statistical modeling. We derive the fundamental properties of the TSLBW distribution, including its probability density function (PDF), cumulative distribution function (CDF), moment-generating function (MGF), and hazard rate function, and establish closed-form expressions for its mean and variance. Parameter estimation is addressed through both the method of moments (MOM) and maximum likelihood estimation (MLE), offering practical tools for real-world applications. Simulation studies and numerical experiments are conducted to evaluate the performance of the proposed distribution and its estimators. The results demonstrate that the TSLBW distribution provides improved flexibility for modeling lifetime data governed by two-sided failure mechanisms, making it a valuable addition to the statistician’s toolkit for reliability inference.