Lifetime Distribution Based on Generators for Discrete Mixtures with Application to Lomax Distribution

In various application areas, data is frequently collected and analyzed using basic statistical distributions such as exponential, Poisson, and gamma distributions. However, these traditional distributions often fail to adequately capture the inherent heterogeneity present in real-world data. This limitation highlights the need for more flexible distributions that can address these complexities. Such distributions can be generated through techniques like reparameterization, generalization, compounding, and mixing. This paper focuses on deriving generators for survival functions of discrete mixtures using minimum and maximum order statistic distributions. The approach leverages the probability generating function (PGF) techniques of mixing distributions, including zero-truncated Poisson, shifted geometric, zero-truncated binomial, zero-truncated negative binomial, and logarithmic series distributions. Specifically, the derived generator was applied to Lomax distributions to construct survival functions. Consequently, the probability density function (PDF) and failure rate of the resulting discrete mixtures were also obtained. Furthermore, the paper examines the shapes of the PDF and failure rate for discrete mixtures derived from the zero-truncated Poisson distribution. Notably, the failure rates of discrete mixtures generated using minimum and maximum order statistics from the Lomax distribution exhibited distinct behaviors. The failure rate for the minimum order statistic was observed to decrease, while the failure rate for the maximum order statistic showed a combination of non-decreasing and bathtub-shaped patterns.