## Bernoulli

• Bernoulli
• Volume 18, Number 2 (2012), 455-475.

### A multivariate piecing-together approach with an application to operational loss data

#### Abstract

The univariate piecing-together approach (PT) fits a univariate generalized Pareto distribution (GPD) to the upper tail of a given distribution function in a continuous manner. We propose a multivariate extension. First it is shown that an arbitrary copula is in the domain of attraction of a multivariate extreme value distribution if and only if its upper tail can be approximated by the upper tail of a multivariate GPD with uniform margins.

The multivariate PT then consists of two steps: The upper tail of a given copula $C$ is cut off and substituted by a multivariate GPD copula in a continuous manner. The result is again a copula. The other step consists of the transformation of each margin of this new copula by a given univariate distribution function.

This provides, altogether, a multivariate distribution function with prescribed margins whose copula coincides in its central part with $C$ and in its upper tail with a GPD copula.

When applied to data, this approach also enables the evaluation of a wide range of rational scenarios for the upper tail of the underlying distribution function in the multivariate case. We apply this approach to operational loss data in order to evaluate the range of operational risk.

#### Article information

Source
Bernoulli, Volume 18, Number 2 (2012), 455-475.

Dates
First available in Project Euclid: 16 April 2012

https://projecteuclid.org/euclid.bj/1334580720

Digital Object Identifier
doi:10.3150/10-BEJ343

Mathematical Reviews number (MathSciNet)
MR2922457

Zentralblatt MATH identifier
1238.62062

#### Citation

Aulbach, Stefan; Bayer, Verena; Falk, Michael. A multivariate piecing-together approach with an application to operational loss data. Bernoulli 18 (2012), no. 2, 455--475. doi:10.3150/10-BEJ343. https://projecteuclid.org/euclid.bj/1334580720

#### References

• [1] Alsina, C., Nelsen, R.B. and Schweizer, B. (1993). On the characterization of a class of binary operations on distribution functions. Statist. Probab. Lett. 17 85–89.
• [2] Balkema, A.A. and de Haan, L. (1974). Residual life time at great age. Ann. Probab. 2 792–804.
• [3] Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. (2004). Statistics of Extremes: Theory and Applications. Chichester: Wiley.
• [4] Berg, D. (2009). Copula goodness-of-fit testing: An overview and power comparison. The European Journal of Finance 15 675–701.
• [5] Biagini, F. and Ulmer, S. (2009). Asymptotics for operational risk quantified with expected shortfall. ASTIN Bulletin 39 735–752.
• [6] Buishand, T.A., de Haan, L. and Zhou, C. (2008). On spatial extremes: With application to a rainfall problem. Ann. Appl. Statist. 2 624–642.
• [7] Cruz, M.G. (2002). Modeling, Measuring and Hedging Operational Risk. Chichester: Wiley.
• [8] D’Agostino, L. and Stephens, M.A. (1986). Goodness-of-fit Techniques. Statistics: Textbooks and Monographs 68. New York: Marcel Dekker Inc.
• [9] de Haan, L. and de Ronde, J. (1998). Sea and wind: Multivariate extremes at work. Extremes 1 7–45.
• [10] de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. New York: Springer.
• [11] de Haan, L. and Sinha, A.K. (1999). Estimating the probability of a rare event. Ann. Statist. 27 732–759.
• [12] Deheuvels, P. (1978). Caractérisation complète des lois extrême multivariées et de la convergence des types extrêmes. Publ. Inst. Statist. Univ. Paris 23 1–36.
• [13] Deheuvels, P. (1984). Probabilistic aspects of multivariate extremes. In Statistical Extremes and Applications (Vimeiro, 1983). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. (J. Tiago de Oliveira, ed.) 131 117–130. Dordrecht: Reidel.
• [14] Di Clemente, A. and Romano, C. (2004). A copula-extreme value theory approach for modelling operational risk. In Operational Risk Modelling and Analysis, Theory and Practice (M. Cruz, ed.) 189–208. London: RISK Books.
• [15] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Applications of Mathematics (New York) 33. Berlin: Springer.
• [16] Falk, M. (2008). It was 30 years ago today when Laurens de Haan went the multivariate way. Extremes 11 55–80.
• [17] Falk, M., Hüsler, J. and Reiss, R.D. (2010). Laws of Small Numbers: Extremes and Rare Events, 3rd ed. Basel: Birkhäuser.
• [18] Falk, M. and Michel, R. (2009). Testing for a multivariate generalized Pareto distribution. Extremes 12 33–51.
• [19] Galambos, J. (1987). The Asymptotic Theory of Extreme Order Statistics, 2nd ed. Melbourne, FL: Robert E. Krieger Publishing Co. Inc.
• [20] Genest, C., Quesada Molina, J.J., Rodríguez Lallena, J.A. and Sempi, C. (1999). A characterization of quasi-copulas. J. Multivariate Anal. 69 193–205.
• [21] Genest, C., Rémillard, B. and Beaudoin, D. (2009). Goodness-of-fit tests for copulas: A review and a power study. Insurance Math. Econom. 44 199–213.
• [22] Huang, X. (1992). Statistics of bivariate extreme values. Ph.D. thesis, Tinbergen Institute Research Series.
• [23] McNeil, A.J., Frey, R. and Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques and Tools. Princeton, NJ: Princeton Univ. Press.
• [24] McNeil, A.J. and Nešlehová, J. (2009). Multivariate Archimedean copulas, d-monotone functions and l1-norm symmetric distributions. Ann. Statist. 37 3059–3097.
• [25] Michel, R. (2006). Simulation and estimation in multivariate generalized Pareto models. Ph.D. thesis, Univ. Würzburg.
• [26] Michel, R. (2007). Simulation of certain multivariate generalized Pareto distributions. Extremes 10 83–107.
• [27] Michel, R. (2008). Some notes on multivariate generalized Pareto distributions. J. Multivariate Anal. 99 1288–1301.
• [28] Moscadelli, M. (2004). The modelling of operational risk: Experience with the analysis of the data collected by the Basel Committee. Banca D’Italia, Termini di discussione No. 517.
• [29] Nelsen, R.B. (2006). An Introduction to Copulas, 2nd ed. New York: Springer.
• [30] Pickands, J. III (1975). Statistical inference using extreme order statistics. Ann. Statist. 3 119–131.
• [31] Rachev, S.T., Stein, M. and Sun, W. (2009). Copula concepts in financial markets. Technical report, Karslruhe Institute of Technology (KIT).
• [32] Reiss, R.D. (1989). Approximate Distributions of Order Statistics: With Applications to Nonparametric Statistics. New York: Springer.
• [33] Reiss, R.D. and Thomas, M. (2007). Statistical Analysis of Extreme Values, 3rd ed. Basel: Birkhäuser.
• [34] Resnick, S.I. (2006). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. New York: Springer.
• [35] Rootzén, H. and Tajvidi, N. (2006). Multivariate generalized Pareto distributions. Bernoulli 12 917–930.