Bayesian Analysis and Markov Chain Monte Carlo Simulation

This chapter describes the use of hierarchical Bayesian analysis using Markov chain Monte Carlo (MCMC) estimation to obtain the parameters of a peaks-over-threshold (POT) extreme value theory model of extreme operational losses. Principal MCMC techniques are described and a simple example which illustrates the concepts, including economic capital allocation, is discussed.

References

1 Raiffa, H. & Schlaifer, R. (1961). Applied Statistical Decision Theory, Harvard University Press.
Web of Science® Google Scholar
2 Medova, E.A. & Kyriacou, M.N. (2002). Extremes in operational risk measurement, in Risk Management: Value At Risk And Beyond, M.A.H. Dempster, ed, Cambridge University Press, pp. 247–274.
10.1017/CBO9780511615337.010
Google Scholar
3 Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A. & Teller, E. (1953). Equations of state calculations by fast computing machines, Journal of Chemical Physics 21(1), 1087–1092.
10.1063/1.1699114
CAS Google Scholar
4 Hastings, W. (1970). Monte Carlo sampling methods using Markov chains and their applications, Biometrika 57(1), 97–109.
10.1093/biomet/57.1.97
Web of Science® Google Scholar
5 Chib, S. & Greenberg, E. (1995). Understanding the Metropolis-Hastings algorithm, The American Statistician 49(4), 327–335.
10.1080/00031305.1995.10476177
Web of Science® Google Scholar
6 Müller, P. (1993). A generic approach to posterior integration and Gibbs sampling, Technical Report, Purdue University.
Google Scholar
7 Tierney, L. (1994). Markov chains for exploring posterior distributions, Annals of Statistics 22, 1701–1762.
10.1214/aos/1176325750
Web of Science® Google Scholar
8 Dongarra, J. & Sullivan, F. (2000). The top 10 algorithms, Computing in Science and Engineering 2(1), 22–23.
10.1109/MCISE.2000.814652
Web of Science® Google Scholar
9 Beichl, I. & Sullivan, F. (2000). The Metropolis algorithm, Computing in Science and Engineering 2(1), 65–69.
10.1109/5992.814660
Web of Science® Google Scholar
10 Roberts, G., Gelman, A. & Gilks, W. (1994). Weak convergence and optimal scaling of random walk Metropolis algorithms, Technical Report, University of Cambridge.
Google Scholar
11 Casella, G. & Edward, I.G. (1992). Explaining the Gibbs sampler, The American Statistician 46, 167–117.
10.1080/00031305.1992.10475878
Web of Science® Google Scholar
12 Medova, E.A. (2001). Operational risk capital allocation and integration of risks, Advances in Operational Risk: Firmwide Issues for Financial Institutions, Risk Books, pp. 115–127.
Google Scholar
13 Embrechts, P., Kluppelberg, C. & Mikosch, T. (1997). Modelling Extremal Events, Springer, Berlin.
10.1007/978-3-642-33483-2
Google Scholar
14 Galambos, J. (1978). The Asymptotic Theory of Extreme Order Statistics, John Wiley & Sons, New York.
Google Scholar
15 Bank of International Settlements (2001). The Bank of International Settlements, Basel, New Basel Capital Accord, Press Release.
Google Scholar
16 Leadbetter, M., LindGreen, G. & Rootzen, H. (1983). Extremes and Related Properties of Random Sequences and Processes, Springer, Berlin.
10.1007/978-1-4612-5449-2
Google Scholar
17 Leadbetter, M. (1991). On a basis for “Peaks over Threshold” modeling, Statistics and Probability Letters 12, 357–362.
10.1016/0167-7152(91)90107-3
Web of Science® Google Scholar
18 Smith, R. (2001). Measuring risk with extreme value theory, Chapter 8, in Risk Management: Value at Risk and Beyond, M.A.H. Dempster, ed, Cambridge University Press.
Google Scholar
19 Castillo, E. (1988). Extreme Value Theory in Engineering, Academic Press, Orlando.
Google Scholar
20 Smith, A. & Roberts, G. (1993). Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods, Journal of the Royal Statistical Society B55, 3–23.
Google Scholar

Citing Literature

Encyclopedia of Quantitative Risk Analysis and Assessment

Browse other articles of this reference work:

Bayesian Analysis and Markov Chain Monte Carlo Simulation

Abstract

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Bayesian Analysis and Markov Chain Monte Carlo Simulation

Abstract

References

Citing Literature

References

Related

Information