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E-raamat: Marshall Olkin Distributions - Advances in Theory and Applications: Bologna, Italy, October 2013

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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.
1 A Survey of Dynamic Representations and Generalizations of the Marshall--Olkin Distribution
1(14)
German Bernhart
Lexuri Fernandez
Jan-Frederik Mai
Steffen Schenk
Matthias Scherer
1.1 The Classical Construction of the Marshall-Olkin Law
2(3)
1.1.1 The Exogenous Shock Model Is "Static"
4(1)
1.2 Interpretation via Poisson Processes
5(1)
1.2.1 Generalization to Cox Processes
5(1)
1.3 The Iterative Construction of Barry Arnold
6(2)
1.3.1 Generalization to Multivariate Phase-Type Distributions
7(1)
1.4 The Levy-Frailty Construction
8(4)
1.4.1 Generalizations
9(3)
1.5 Conclusion
12(3)
References
12(3)
2 Copulas Based on Marshall--Olkin Machinery
15(18)
Fabrizio Durante
Stephane Girard
Gildas Mazo
2.1 Introduction
15(1)
2.2 Marshall--Olkin Machinery
16(1)
2.3 Copulas Generated by One Independent Shock
17(7)
2.3.1 The Bivariate Case
19(3)
2.3.2 The Multivariate Case
22(2)
2.4 Combining Marshall--Olkin Bivariate Copulas to Get Flexible Multivariate Models
24(3)
2.5 Some Comments About Statistical Inference Procedures
27(2)
2.6 Conclusions
29(4)
References
30(3)
3 The Mean of Marshall--Olkin-Dependent Exponential Random Variables
33(18)
Lexuri Fernandez
Jan-Frederik Mai
Matthias Scherer
3.1 Introduction
33(1)
3.2 The Marshall--Olkin Law
34(4)
3.3 The Exchangeable Marshall--Olkin Law
38(8)
3.4 The Extendible Marshall--Olkin Law
46(3)
3.5 Conclusion
49(2)
References
49(2)
4 General Marshall--Olkin Models, Dependence Orders, and Comparisons of Environmental Processes
51(14)
Esther Frostig
Franco Pellerey
4.1 Introduction
51(1)
4.2 Preliminaries
52(4)
4.2.1 Dependence Orders
53(2)
4.2.2 Some Univariate Stochastic Orders
55(1)
4.2.3 The General Model for the Environmental Processes
55(1)
4.3 Comparison Results
56(2)
4.4 Supermodular Comparison of Environmental Processes
58(7)
4.4.1 An Application in Population Dynamics
58(2)
4.4.2 An Application in Comparisons of Collective Risks
60(1)
4.4.3 An Application in Ruin Theory
61(1)
4.4.4 An Application in Reliability
62(1)
References
63(2)
5 Marshall--Olkin Machinery and Power Mixing: The Mixed Generalized Marshall--Olkin Distribution
65(22)
Sabrina Mulinacci
5.1 Introduction
65(2)
5.2 Mixtures of Survival Distributions on [ 0, +∞)d
67(6)
5.2.1 Min-ID Distributions
67(3)
5.2.2 Power-Mixtures of Survival Min-ID Distributions
70(1)
5.2.3 Conditional Mixed Survival Min-ID Distributions
71(2)
5.3 Power-Mixture Closure Under the MO-machinery
73(4)
5.3.1 The General MO-machinery
73(2)
5.3.2 Min-ID Closure Under the MO-machinery
75(1)
5.3.3 Power-Mixture Closure Under the MO-machinery
76(1)
5.4 The Generalized Marshall--Olkin Type Distributions
77(4)
5.5 The Mixed Generalized Marshall--Olkin Distribution
81(6)
References
86(1)
6 Extended Marshall--Olkin Model and Its Dual Version
87
Jayme Pinto
Nikolai Kolev
6.1 Introduction
87(2)
6.2 Extended Marshall--Olkin Model
89(9)
6.2.1 Alternative Representation of the EMO Model
90(1)
6.2.2 Probabilistic Properties
91(2)
6.2.3 Survival Copula of the EMO Model
93(3)
6.2.4 Distributional Property for Residual Lifetimes
96(2)
6.3 Bayesian Data Analysis with EMO Models
98(6)
6.4 A Dual Version of EMO Model
104(6)
6.4.1 Model Specification and Basic Probabilistic Properties
105(3)
6.4.2 Stochastic Order Comparisons
108(2)
6.5 Concluding Remarks
110
References
112
Umberto Cherubini is an Associate Professor of Financial Mathematics at the University of Bologna. He is member of the scientific committee of AIFIRM, the Italian Association of Financial Risk Managers, and of ABI Formazione, the education branch of the Italian Banking Association. He has been consulting and teaching in the field of finance and risk management, with particular focus on multivariate products and dependence, for more than ten years. Before joining the academia, he worked at the Economic Research Department of BCI (COMIT), an Italian bank, where he directed the Forecasting and Risk Measurement Methods Unit. He has published papers in finance and economics in international journasl and six books on topics of risk management and financial mathematics.

Fabrizio Durante is an Associate Professor for Statistics at the Faculty of Economics and Management of the Free University of Bozen-Bolzano (UNIBZ), Italy. He studied at the University of Lecce, Italy, where he has obtained his doctoral degree in Mathematics. Before joining FUB, he was also working at the Johannes Kepler University Linz, Austria, where he has obtained the habilitation in Mathematics in 2010.  His research activities focus on the fields of dependence and copula models, with particular emphasis on applications in financial risk management, reliability theory, and environmental science (especially hydrology). He co-edited two books devoted to copula theory and its applications published by Springer. Moreover, he has organized several international events and special sessions about dependence models in the recent years. Currently, he is associate editor of the journal Computational Statistics and Data Analysis" and Dependence Modeling".

Sabrina Mulinacci is an Associate Professor of Mathematical Methods for Economics and Finance at the Department of Statistics at the University of Bologna. She obtained her PhD inmathematics at the University of Pisa and she has been Associate Professor at the Catholic University of Milan.  Her main research interests focus on probability theory and mathematical finance.
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