Marshall–Olkin X–Exponential Distribution with an Autoregressive Minification Process: Theory, Inference, and Application

Abstract

In this paper, we introduce a two-parameter lifetime distribution called the Marshall – Olkin X – Exponential distribution (MOXED). The model is obtained by applying the Marshall – Olkin shock transformation to the X – Exponential distribution. The proposed distribution contains the X – Exponential model as a special case and adds a Marshall – Olkin parameter that improves flexibility, especially for data with early failures and right-skewed behavior. Explicit expressions are derived for the probability density function, distribution function, survival function, and hazard rate function. The reliability behavior of the model is also examined. In particular, although the baseline X – Exponential distribution has an increasing hazard rate, the Marshall – Olkin extension can generate initially decreasing and possibly non-monotone hazard-rate patterns, depending on the value of the additional parameter. Moment expressions are obtained using both integral and series representations. We also develop a first-order minification autoregressive process, denoted by MOXED – AR(1), whose stationary marginal distribution is MOXED. The construction is based on a refresh-and-minification mechanism and provides a natural interpretation for positive-valued dependent observations. Stationarity, transition behavior, conditional moments, dependence properties, and geometric ergodicity are established. Maximum likelihood estimation is considered, and a Monte Carlo simulation study is carried out to examine the finite-sample performance of the estimators. Finally, a real failure-time data set is analyzed and compared with several classical lifetime models. The empirical application focuses on the marginal MOXED distribution, while the MOXED – AR(1) process provides a framework for future modeling of dependent positive-valued sequences. The results show that the MOXED model provides a competitive and interpretable fit, particularly in terms of likelihood-based criteria, without claiming uniform superiority over all competing models and all goodness-of-fit measures.