Recent Developments About Marshall-Olkin Bivariate Distribution

Abstract

This paper is a short review of classical and recent results on Marshall-Olkin shock models and their applications in reliability analysis. The classical Marshall-Olkin shock model was introduced in Marshall and Olkin (J Am Stat Assoc 62:30-44, 1967). The model describes a joint distribution of lifetimes of two components of a system subjected to three types of shocks. The distribution has absolutely continuous and singular parts. The Marshall-Olkin copula also aroused the interest of researchers working on the theory of copulas as an example of a copula having absolutely continuous and singular parts. There are some recent papers considering general models and modifications constructed on the basic idea of Marshall and Olkin (1967). These works find wide applications in reliability analysis in the case of a general system having n (n > 2) components and shocks coming from in (m > 3) sources. Some applications can also be seen in the theory of credit risk, where instead of lifetimes of the components, one considers the times to the default of two counter-parties subject to three independent underlying economic or financial events. In this work, we analyze and describe the results dealing with the generalization and modification of the Marshall-Olkin model.