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Recent Developments About Marshall–Olkin Bivariate Distribution

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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 m (\(m > 3\)) 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.

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Acknowledgements

The authors thank the anonymous reviewer for valuable comments and suggestions that led to improvements in the paper.

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  1. Department of Mathematics, Izmir University of Economics, Izmir, Turkey

    Ismihan Bayramoglu & Murat Ozkut

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  1. Ismihan Bayramoglu
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  2. Murat Ozkut
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Correspondence to Ismihan Bayramoglu.

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This work is devoted to the memory of Professor Theofilos Cacoullos.

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This article is part of the topical collection “Advances in Probability and Statistics: an Issue in Memory of Theophilos Cacoullos” guest edited by Narayanaswamy Balakrishnan, Charalambos A. Charalambides, Tasos Christofides, Markos Koutras, and Simos Meintanis.

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Bayramoglu, I., Ozkut, M. Recent Developments About Marshall–Olkin Bivariate Distribution. J Stat Theory Pract 16, 58 (2022). https://doi.org/10.1007/s42519-022-00278-4

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