Some New Contributions on the Marshall–Olkin Length Biased Lomax Distribution: Theory, Modelling and Data Analysis

Abstract

The Lomax distribution is arguably one of the most useful lifetime distributions, explaining the developments of its extensions or generalizations through various schemes. The Marshall–Olkin length-biased Lomax distribution is one of these extensions. The associated model has been used in the frameworks of data fitting and reliability tests with success. However, the theory behind this distribution is non-existent and the results obtained on the fit of data were sufficiently encouraging to warrant further exploration, with broader comparisons with existing models. This study contributes in these directions. Our theoretical contributions on the the Marshall–Olkin length-biased Lomax distribution include an original compounding property, various stochastic ordering results, equivalences of the main functions at the boundaries, a new quantile analysis, the expressions of the incomplete moments under the form of a series expansion and the determination of the stress–strength parameter in a particular case. Subsequently, we contribute to the applicability of the Marshall–Olkin length-biased Lomax model. When combined with the maximum likelihood approach, the model is very effective. We confirm this claim through a complete simulation study. Then, four selected real life data sets were analyzed to illustrate the importance and flexibility of the model. Especially, based on well-established standard statistical criteria, we show that it outperforms six strong competitors, including some extended Lomax models, when applied to these data sets. To our knowledge, such comprehensive applied work has never been carried out for this model.