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A Survey of Dynamic Representations and Generalizations of the Marshall–Olkin Distribution

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Marshall ̶ Olkin Distributions - Advances in Theory and Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 141))

Abstract

In the classical stochastic representation of the Marshall–Olkin distribution, the components are interpreted as future failure times which are defined as the minimum of independent, exponential arrival times of exogenous shocks. Many applications only require knowledge about the failure times before a given time horizon, i.e. the model is “truncated” at a fixed maturity. Unfortunately, such a truncation is infeasible with the original exogenous shock model, because it is a priori unknown which arrival times of exogenous shocks are relevant and which ones occur after the given time horizon. In this sense, the original model lacks a time-dynamic nature. Fortunately, the characterization in terms of the lack-of-memory property gives rise to several alternative stochastic representations which are consistent with a dynamic viewpoint in the sense that a stochastic simulation works along a time line and can thus be stopped at an arbitrary horizon. Building upon this dynamic viewpoint, some of the alternative representations lead to interesting generalizations of the Marshall–Olkin distribution. The present article surveys the literature in this regard.

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Notes

  1. 1.

    \(\fancyscript{P}(\{1, \ldots , d\})\) means the power set of \(\{1, \ldots , d\}\).

  2. 2.

    See [25], Chap. 5, for these results.

  3. 3.

    See [41] for details on Bernstein functions.

  4. 4.

    See [40] for background on Lévy subordinators.

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Authors and Affiliations

  1. Lehrstuhl für Finanzmathematik (M13), Technische Universität München, Parkring 11, 85748, Garching-Hochbrück, Germany

    German Bernhart, Lexuri Fernández, Steffen Schenk & Matthias Scherer

  2. XAIA Investment, Sonnenstr. 19, 80331, München, Germany

    Jan-Frederik Mai

Authors
  1. German Bernhart
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  2. Lexuri Fernández
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  3. Jan-Frederik Mai
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  4. Steffen Schenk
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  5. Matthias Scherer
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Correspondence to Matthias Scherer .

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Editors and Affiliations

  1. Department of Statistics, University of Bologna, Bologna, Italy

    Umberto Cherubini

  2. Free University of Bozen-Bolzano, Faculty of Economics and Management, Bolzano, Italy

    Fabrizio Durante

  3. University of Bologna, Department of Statistics, Bologna, Italy

    Sabrina Mulinacci

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Bernhart, G., Fernández, L., Mai, JF., Schenk, S., Scherer, M. (2015). A Survey of Dynamic Representations and Generalizations of the Marshall–Olkin Distribution. In: Cherubini, U., Durante, F., Mulinacci, S. (eds) Marshall ̶ Olkin Distributions - Advances in Theory and Applications. Springer Proceedings in Mathematics & Statistics, vol 141. Springer, Cham. https://doi.org/10.1007/978-3-319-19039-6_1

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