Marshall-Olkin Poisson-Exponentiated Weibull Distribution: Properties and Applications

In this paper, we introduce a new five-parameter lifetime distribution, called the Marshall-Olkin Poisson Exponentiated Weibull (MOPEW) distribution. The proposed model is constructed by combining the Poisson-X family with the exponentiated Weibulldistribution and subsequently applying the Marshall-Olkin transformation. The resulting distribution exhibits substantialflexibility in modeling various shapes of density and hazard rate functions, including increasing, decreasing, bathtub, and unimodalforms. We investigate several structural properties of the proposed distribution, including explicit expressions for the density,survival, and hazard functions, as well as moments and quantile function. Parameter estimation is performed via the maximumlikelihood method, and the asymptotic properties of the estimators are discussed. It is shown that the proposed model provides enhanced flexibility in capturing variability, skewness, and tail behavior in continuous lifetime data, compared to classical Weibull-type models. The usefulness of the MOPEW distribution is illustrated through applications to real datasets, where it demonstrates improved goodness-of-fit according to standard model selection criteria. The proposed model substantially improves theflexibility of PEW by allowing additional control over tail heaviness, skewness, and hazard-rate shapes. We derive itsmathematical properties, obtain maximum likelihood estimators, and study their performance through extensive simulations. Applications to tax-revenue data and to AR(1) time series modeling (via MIN and MAX-MIN structures) demonstrate that MOPEW provides superior fit and forecasting accuracy compared with PEW and other Marshall-Olkin variants. The distribution therefore offers a versatile tool for over- and under dispersed data and to show the feasibility of the proposed distribution in modeling AR(1) time series data and the COVID-19 death rates of USA is considered.