## Bernoulli

• Bernoulli
• Volume 17, Number 1 (2011), 226-252.

### Conditioning on an extreme component: Model consistency with regular variation on cones

#### Abstract

Multivariate extreme value theory assumes a multivariate domain of attraction condition for the distribution of a random vector. This necessitates that each component satisfies a marginal domain of attraction condition. An approximation of the joint distribution of a random vector obtained by conditioning on one of the components being extreme was developed by Heffernan and Tawn [12] and further studied by Heffernan and Resnick [11]. These papers left unresolved the consistency of different models obtained by conditioning on different components being extreme and we here provide clarification of this issue. We also clarify the relationship between these conditional distributions, multivariate extreme value theory and standard regular variation on cones of the form $[0, ∞]×(0, ∞]$.

#### Article information

Source
Bernoulli, Volume 17, Number 1 (2011), 226-252.

Dates
First available in Project Euclid: 8 February 2011

https://projecteuclid.org/euclid.bj/1297173841

Digital Object Identifier
doi:10.3150/10-BEJ271

Mathematical Reviews number (MathSciNet)
MR2797990

Zentralblatt MATH identifier
1284.60103

#### Citation

Das, Bikramjit; Resnick, Sidney I. Conditioning on an extreme component: Model consistency with regular variation on cones. Bernoulli 17 (2011), no. 1, 226--252. doi:10.3150/10-BEJ271. https://projecteuclid.org/euclid.bj/1297173841

#### References

• [1] Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987). Regular Variation. Cambridge: Cambridge University Press.
• [2] Das, B. (2009). The conditional extreme value model and related topics. Ph.D. thesis, Cornell University, Ithaca, NY.
• [3] Das, B. and Resnick, S.I. (2009). Detecting a conditional extreme value model. Extremes DOI: 10.1007/s10687-009-0097-3. Available at http://arxiv.org/abs/0902.2996.
• [4] Davydov, Y., Molchanov, I. and Zuyev, S. (2007). Stable distributions and harmonic analysis on convex cones. C. R. Math. Acad. Sci. Paris 344 321–326.
• [5] de Haan, L. (1978). A characterization of multidimensional extreme-value distributions. Sankhyā Ser. A 40 85–88.
• [6] de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. New York: Springer.
• [7] de Haan, L. and Resnick, S.I. (1977). Limit theory for multivariate sample extremes. Z. Wahrsch. Verw. Gebiete 40 317–337.
• [8] de Haan, L. and Resnick, S.I. (1979). Conjugate π-variation and process inversion. Ann. Probab. 7 1028–1035.
• [9] Fougères, A. and Soulier, P. (2009). Estimation of conditional laws given an extreme component. Available at http://arXiv.org/0806.2426v2.
• [10] Geluk, J.L. and de Haan, L. (1987). Regular Variation, Extensions and Tauberian Theorems. CWI Tract 40. Amsterdam: Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica.
• [11] Heffernan, J.E. and Resnick, S.I. (2007). Limit laws for random vectors with an extreme component. Ann. Appl. Probab. 17 537–571.
• [12] Heffernan, J.E. and Tawn, J.A. (2004). A conditional approach for multivariate extreme values (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 66 497–546.
• [13] Kallenberg, O. (1983). Random Measures, 3rd edition. Berlin: Akademie.
• [14] Klüppelberg, C. and Resnick, S.I. (2008). The Pareto copula, aggregation of risks and the emperor’s socks. J. Appl. Probab. 45 67–84.
• [15] Ledford, A.W. and Tawn, J.A. (1996). Statistics for near independence in multivariate extreme values. Biometrika 83 169–187.
• [16] Ledford, A.W. and Tawn, J.A. (1997). Modelling dependence within joint tail regions. J. Roy. Statist. Soc. Ser. B 59 475–499.
• [17] López-Oliveros, L. and Resnick, S.I. (2009). Extremal dependence analysis of network sessions. Extremes DOI: 10.1007/s10687-009-0096-4. Available at http://arxiv.org/pdf/0905.1983v1.
• [18] Maulik, K. and Resnick, S.I. (2005). Characterizations and examples of hidden regular variation. Extremes 7 31–67.
• [19] Mikosch, T. (2005). How to model multivariate extremes if one must? Statist. Neerlandica 59 324–338.
• [20] Mikosch, T. (2006). Copulas: Tales and facts. Extremes 9 3–20.
• [21] Mitra, A. and Resnick, S.I. (2010). Hidden regular variation: Detection and estimation. Available at http://people.orie.cornell.edu/~sid.
• [22] Neveu, J. (1977). Processus ponctuels. In École d’Été de Probabilités de Saint-Flour, VI–1976. Lecture Notes in Math. 598 249–445. Berlin: Springer.
• [23] Pickands, J. (1975). Statistical inference using extreme order statistics. Ann. Statist. 3 119–131.
• [24] Pickands, J. (1981). Multivariate extreme value distributions. In 43rd Sess. Int. Statist. Inst. Buenos Aires 859–878.
• [25] Ramos, A. and Ledford, A.W. (2009). A new class of models for bivariate joint tails. J. Roy. Statist. Soc. Ser. B 71.
• [26] Resnick, S.I. (2002). Hidden regular variation, second order regular variation and asymptotic independence. Extremes 5 303–336.
• [27] Resnick, S.I. (2007). Heavy Tail Phenomena: Probabilistic and Statistical Modeling. New York: Springer.
• [28] Resnick, S.I. (2008). Multivariate regular variation on cones: Application to extreme values, hidden regular variation and conditioned limit laws. Stochastics 80 269–298.
• [29] Resnick, S.I. (2008). Extreme Values, Regular Variation and Point Processes. New York: Springer.
• [30] Resnick, S.I. and Zeber, D. (2010). Foundations of conditional extreme value theory. To appear.
• [31] Schlather, M. (2001). Examples for the coefficient of tail dependence and the domain of attraction of a bivariate extreme value distribution. Statist. Probab. Lett. 53 325–329.
• [32] Seneta, E. (1976). Regularly Varying Functions. Lecture Notes in Math. 508. New York: Springer.