The Weibull-generalized shifted geometric distribution: properties, estimation, and applications

This paper introduces the Weibull-generalized shifted geometric (WGSG) distribution, a novel lifetime model integrating the Weibull and shifted geometric distributions to address complex lifetime data patterns. Extending the Weibull-geometric framework, this distribution models system reliability by focusing on the $k$-th smallest lifetime—when $k$ components fail—rather than the minimum. Key properties, including the probability density function, cumulative distribution function, and moments, were derived. Parameters were estimated using maximum likelihood, expectation-maximization, method of moments, and Bayesian approaches, with a simulation study comparing their performance. Applications to two real-world lifetime and reliability datasets demonstrated the distribution's superiority over classical models in handling challenging survival and reliability scenarios. This flexible model enhances the ability to capture diverse hazard behaviors, advancing lifetime data analysis.