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## Über dieses Buch

This book presents the latest advances in the theory and practice of Marshall-Olkin distributions. These distributions have been increasingly applied in statistical practice in recent years, as they make it possible to describe interesting features of stochastic models like non-exchangeability, tail dependencies and the presence of a singular component. The book presents cutting-edge contributions in this research area, with a particular emphasis on financial and economic applications. It is recommended for researchers working in applied probability and statistics, as well as for practitioners interested in the use of stochastic models in economics. This volume collects selected contributions from the conference “Marshall-Olkin Distributions: Advances in Theory and Applications,” held in Bologna on October 2-3, 2013.

## Inhaltsverzeichnis

### Chapter 1. A Survey of Dynamic Representations and Generalizations of the Marshall–Olkin Distribution

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.
German Bernhart, Lexuri Fernández, Jan-Frederik Mai, Steffen Schenk, Matthias Scherer

### Chapter 2. Copulas Based on Marshall–Olkin Machinery

Abstract
We present a general construction principle for copulas that is inspired by the celebrated Marshall–Olkin exponential model. From this general construction method, we derive special subclasses of copulas that could be useful in different situations and recall their main properties. Moreover, we discuss possible estimation strategy for the proposed copulas. The presented results are expected to be useful in the construction of stochastic models for lifetimes (e.g., in reliability theory) or in credit risk models.
Fabrizio Durante, Stéphane Girard, Gildas Mazo

### Chapter 3. The Mean of Marshall–Olkin-Dependent Exponential Random Variables

Abstract
The probability distribution of $$S_d:=X_1+\cdots +X_d$$, where the vector $$(X_1,\ldots ,X_d)$$ is distributed according to the Marshall–Olkin law, is investigated. Closed-form solutions are derived in the general bivariate case and for $$d\in \{2,3,4\}$$ in the exchangeable subfamily. Our computations can, in principle, be extended to higher dimensions, which, however, becomes cumbersome due to the large number of involved parameters. For the Marshall–Olkin distributions with conditionally independent and identically distributed components, however, the limiting distribution of $$S_d/d$$ is identified as $$d$$ tends to infinity. This result might serve as a convenient approximation in high-dimensional situations. Possible fields of application for the presented results are reliability theory, insurance, and credit-risk modeling.
Lexuri Fernández, Jan-Frederik Mai, Matthias Scherer

### Chapter 4. General Marshall–Olkin Models, Dependence Orders, and Comparisons of Environmental Processes

Abstract
In many applicative fields, the behavior of a process $$\mathbf {Z}$$ is assumed to be subjected to an underlying process $$\varTheta$$ that describes evolutions of environmental conditions. A common way to define the environmental process is by letting the marginal values of $$\varTheta$$ subjected to specific environmental factors (constant along time) and factors describing the conditions of the environment at the specified time. In this paper we describe some recent results that can be used to compare two of such environmental discrete-time processes $$\varTheta$$ and $$\widetilde{\varTheta }$$ in dependence. A sample of applications of the effects of these comparison results on the corresponding processes $$\mathbf {Z}$$ and $$\widetilde{\mathbf {Z}}$$ in some different applicative contexts are provided.
Esther Frostig, Franco Pellerey

### Chapter 5. Marshall–Olkin Machinery and Power Mixing: The Mixed Generalized Marshall–Olkin Distribution

Abstract
In this paper, we consider the Marshall–Olkin technique of modeling the multivariate random lifetimes of the components of a system, as the first arrival times of some shock affecting part or the whole system and we analyze the possibility to add more dependence among the shocks and, as a consequence, among the lifetimes, through the power-mixing technique. This approach is applied to obtain extensions of the generalized Marshall–Olkin distributions.
Sabrina Mulinacci

### Chapter 6. Extended Marshall–Olkin Model and Its Dual Version

Abstract
We propose an extension of the generalized bivariate Marshall–Olkin model assuming dependence between the random variables involved. Probabilistic, aging properties, and survival copula representation of the extended model are obtained and illustrated by examples. Bayesian analysis is performed and possible applications are discussed. A dual version of extended Marshall–Olkin model is introduced and related stochastic order comparisons are presented.
Jayme Pinto, Nikolai Kolev
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