Bayesian and Classical Approaches for Reliability Modeling of Microcircuit Failure Times Using a Lambda Power Exponential Type Distribution

Abstract

This study introduces a new three-parameter lifetime distribution, the Lambda Power Exponential Type (LPET) distribution derived via Lehmann type-I exponentiation of an exponential Extension baseline using power transformation.  A frailty-like shape parameter (α) and a power-based transformation is introduced. The addition of a shape parameter enhances the model flexibility to capture diverse hazard shapes such as decreasing, increasing, bathtub, j-shaped and inverted j- shaped. Both classical maximum likelihood estimation (MLE) and Bayesian analysis using Hamiltonian Monte Carlo (HMC) are employed for the parameter estimation. A Monte Carlo simulation demonstrates a very consistent reduction of both biases, mean square error, and root mean squared error with increasing sample size. Application to real-world microcircuit electromigration failure data verifies the superiority of the LPET model compared to eleven competing lifetime models based on information criteria and goodness of fit tests. Bayesian inference confirms stability of the parameter estimates and aligns closely with Kaplan-Meier survival curve, with a mean time to failure of 7.15 hours and median time to failure of 7.08 hours. The LPET model emerges as a highly flexible and robust model for reliability engineering, survival analysis and time - to-event modeling.