Skip to main content
SpringerLink
Log in

Extreme Value Theory

  • Chapter
  • First Online:
Laws of Small Numbers: Extremes and Rare Events

Abstract

In this chapter we summarize results in extreme value theory, which are primarily based on the condition that the upper tail of the underlying df is in the δ-neighborhood of a generalized Pareto distribution (GPD). This condition, which looks a bit restrictive at first sight (see Section 2.2), is however essentially equivalent to the condition that rates of convergence in extreme value theory are at least of algebraic order (see Theorem 2.2.5). The δ-neighborhood is therefore a natural candidate to be considered, if one is interested in reasonable rates of convergence of the functional laws of small numbers in extreme value theory (Theorem 2.3.2) as well as of parameter estimators (Theorems 2.4.4, 2.4.5 and 2.5.4).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. Balkema, A.A., and Haan, L. de (1972). On R. von Mises’ condition for the domain of attraction of exp(− e −x). Ann. Math. Statist. 43, 1352-1354.

    Article  MATH  MathSciNet  Google Scholar 

  2. Balkema, A.A., and Haan, L. de (1974). Residual life time at great age. Ann. Probab. 2, 792-804.

    Article  MATH  Google Scholar 

  3. Beirlant, J., Goegebeur, Y., Segers, J., and Teugels, J. (2004). Statistics of Extremes. Theory and Applications. Wiley Series in Probability and Statistics, Wiley, Chichester.

    Google Scholar 

  4. Beirlant, J. and Teugels, J. L. T. (1989). Asymptotic normality of Hill’s estimator, In Extreme Value Theory Proceedings (Oberwolfach, 1987, Lecture Notes in Statistics), Vol.51, Springer, New York.

    Google Scholar 

  5. Bingham, N.H., Goldie, C.M., and Teugels, J.L. (1987). Regular Variation. Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  6. Castillo, E., Galambos, J., and Sarabia, J.M. (1989). The selection of the domain of attraction of an extreme value distribution from a set of data. In Extreme Value Theory (J. Hüsler and R.-D. Reiss, eds.), Lecture Notes in Statistics, Springer, New York, 51, 181-190.

    Google Scholar 

  7. Christoph, G., and Falk, M. (1996). A note on domains of attraction of p-max stable laws. Statist. Probab. Letters 28, 279-284.

    Article  MATH  MathSciNet  Google Scholar 

  8. Coles, S.G. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer Series in Statistics, Springer, New York.

    Google Scholar 

  9. Cormann, U., and Reiss, R.-D. (2009). Generalizing the Pareto to the log- Pareto model and statistical inference. Extremes 12, 93-105.

    Article  MathSciNet  Google Scholar 

  10. Cormann, U. (2010). Statistical models for exceedances with applications to finance and environmental statistics. PhD Thesis, Dept. Math., University of Siegen.

    Google Scholar 

  11. Csörgő, S., and Mason, D.M. (1985). Central limit theorems for sums of extreme values. Mathematical Proceedings of the Cambridge Philosophical Society, 98, 547-558.

    Article  MathSciNet  Google Scholar 

  12. Csörgő, S., and Mason, D.M. (1989). Simple estimators of the endpoint of a distribution. In Extreme Value Theory (J. Hüsler and R.-D. Reiss eds.), Lecture Notes in Statistics, Springer, New York, bf 51, 132-147.

    Google Scholar 

  13. Davis, R.A., and Resnick, S.I. (1984). Tail estimates motivated by extreme value theory. Ann. Statist. 12, 1467-1487.

    Article  MATH  MathSciNet  Google Scholar 

  14. Davison, A.C., and Smith, R.L. (1990). Models for exceedances over high thresholds (with discussions). J.. Roy. Statist. Soc., Ser. B 52, 393-442.

    MATH  MathSciNet  Google Scholar 

  15. Desgagné, A., and Angers, J.-F. (2005). Importance sampling with the generalized exponential power density. Statist. Comp. 15, 189-195.

    Article  Google Scholar 

  16. Dekkers, A.L.M., Einmahl, J.H.J., and Haan, L. de (1989). A moment estimator for the index of an extreme-value distribution. Ann. Statist. 17, 1833-1855.

    Article  MATH  MathSciNet  Google Scholar 

  17. Dekkers, A.L.M., and Haan, L. de (1989). On the estimation of the extremevalue index and large quantile estimation. Ann. Statist. 17, 1795-1832.

    Article  MATH  MathSciNet  Google Scholar 

  18. Diebolt, J., El-Aroui, M.-A., Garrido, M., and Girard, S. (2003). Quasiconjugate Bayes estimates for GPD parameters and application to heavy tails modelling. Technical Report.

    Google Scholar 

  19. Diebolt, J., El-Aroui, M.-A., Garrido, M., and Girard, S. (2005). Quasiconjugate Bayes estimates for GPD parameters and application to heavy tails modelling. Extremes 8, 57-78.

    Article  MATH  MathSciNet  Google Scholar 

  20. Drees, H. (1995). Refined Pickands estimator of the extreme value index. Ann. Statist. 23, 2059-2080.

    Article  MATH  MathSciNet  Google Scholar 

  21. Dwass, M. (1966). Extremal processes II. Illinois J. Math. 10, 381-391.

    MATH  MathSciNet  Google Scholar 

  22. Embrechts, P., Klüppelberg, C., and Mikosch, T. (1997). Modelling Extremal Events. Applications of Mathematics, Springer, New York.

    Google Scholar 

  23. Falk, M. (1986). Rates of uniform convergence of extreme order statistics. Ann. Inst. Statist. Math. 38, 245-262.

    Article  MATH  MathSciNet  Google Scholar 

  24. Falk, M. (1990). A note on generalized Pareto distributions and the k upper extremes. Probab. Th. Rel. Fields 85, 499-503.

    Article  MATH  MathSciNet  Google Scholar 

  25. Falk, M. (1995 a). LAN of extreme order statistics. Ann. Inst. Statist. Math. 47, 693-717

    Article  MATH  MathSciNet  Google Scholar 

  26. Falk, M. (1995 b). Some best parameter estimates for distributions with finite endpoint. Statistics 27, 115-125.

    Article  MATH  MathSciNet  Google Scholar 

  27. Falk, M., and Marohn, F. (1993 b). Von Mises conditions revisited. Ann. Probab. 21, 1310-1328.

    Article  MATH  MathSciNet  Google Scholar 

  28. Feller,W. (1971). An Introduction to Probability Theory and Its Applications. Vol II, 2nd. ed., John Wiley, New York.

    MATH  Google Scholar 

  29. Fraga Alves, M.I., Haan, L. de, and Neves, C. (2009). Statistical Inference for heavy and super-heavy tailed distributions. J. Stat. Plan. Inf. 139, 213-227.

    Article  MATH  Google Scholar 

  30. Galambos, J. (1978). The Asymptotic Theory of Extreme Order Statistics. 1st edition, Krieger, Malabar.

    MATH  Google Scholar 

  31. Galambos, J. (1987). The Asymptotic Theory of Extreme Order Statistics. 2nd edition, Krieger, Malabar.

    MATH  Google Scholar 

  32. Geluk, J. L. and de Haan, L. (1987). Regular Variation, Extensions and Tauberian theorems, CWI Tract 40, Center for Mathematics and Computer Science, Amsterdam, the Netherlands.

    Google Scholar 

  33. Geluk, J. L. and de Haan, L. (2000). Stable probability distributions and their domains of attraction: a direct approach, Probability and Mathematical Statistics, 20, 169–188.

    MATH  Google Scholar 

  34. Gnedenko, B. (1943). Sur la distribution limite du terme maximum d’une série aléatoire. Ann. Math. 44, 423-453.

    Article  MathSciNet  Google Scholar 

  35. Haan, L. de (1970). On Regular Variation and its Application to the Weak Convergence of Sample Extremes. Math. Centre Tracts 32, Amsterdam.

    Google Scholar 

  36. Haan, L. de (1971). A form of regular variation and its application to the domain of attraction of the double exponential distriubtion. Z. Wahrsch. Verw. Geb. 17, 241-258.

    Article  MATH  Google Scholar 

  37. Haan, L. de (1988). Statistical work related to the height of sea-dikes. Lecture given at the 18th European Meeting of Statisticians, Berlin, GDR, August 22-26.

    Google Scholar 

  38. Haan, L. de, and Ferreira, A. (2006). Extreme Value Theory. (An Introduction). Springer Series in Operations Research and Financial Engineering, Springer, New York.

    Google Scholar 

  39. Hall, P. (1982 a). On estimating the endpoint of a distribution. Ann. Statist. 10, 556-568.

    Article  MATH  MathSciNet  Google Scholar 

  40. Hall, P. (1982 b). On some simple estimates of an exponent of regular variation. J. Royal Statist. Soc. B 44, 37-42.

    MATH  Google Scholar 

  41. Hall, P., and Welsh, A.H. (1985). Adaptive estimates of parameters of regular variation. Ann. Statist. 13, 331-341.

    Article  MATH  MathSciNet  Google Scholar 

  42. Häusler, E., and Teugels, J.L. (1985). On asymptotic normality of Hill’s estimator for the exponent of regular variation. Ann. Statist. 13, 743-756.

    Article  MATH  MathSciNet  Google Scholar 

  43. Hill, B.M. (1975). A simple approach to inference about the tail of a distribution. Ann. Statist. 3, 1163-1174.

    Article  MATH  MathSciNet  Google Scholar 

  44. Hosking, J.R.M. and Wallis, J.R. (1987). Parameter and quantile estimation for the generalized Pareto distribution. Technometrics, 29, 339-349.

    Article  MATH  MathSciNet  Google Scholar 

  45. Kaufmann, E., and Reiss, R.-D. (2002). An upper bound on the binomial process approximation to the exceedance process. Extremes 5, 253-269.

    Article  MathSciNet  Google Scholar 

  46. Kotz, S., and Nadarajah, S. (2000). Extreme Value Distributions. Theory and Applications. Imperial College Press, London.

    Book  MATH  Google Scholar 

  47. Kremer, E. (1986). Simple formulas for the premiums of the LCR and ECOMOR treaties under exponential claim sizes. Blätter der DGVM, Band XVII, 237-243.

    Article  Google Scholar 

  48. Marron, J.S. (1988). Automatic smoothing parameter selection: A survey. Empirical Economics 13, 187-208.

    Article  Google Scholar 

  49. Meerschaert, M.M., and Scheffler, H.-P. (2006). Stochastic model for ultraslow diffusion. Stoch. Process. Appl. 116, 1215-1235.

    Article  MATH  MathSciNet  Google Scholar 

  50. Mises, R. von (1936). La distribution de la plus grande de n valeurs. Reprinted in Selected papers II. Amer. Math. Soc., Providence, R.I., 1954, 271-294.

    Google Scholar 

  51. Mohan, N.R., and Ravi, S. (1993). Max domains of attraction of univariate and multivariate p-max stable laws. Theory of Probability and its Applications 37, 632-643.

    Article  MathSciNet  Google Scholar 

  52. Mohan, N.R. and Subramanya, U.R. (1991). Characterization of max domains of attraction of univariate p-max stable laws. Proceedings of the Symposium on Distribution Theory, Kochi, Kerala, India, 11-24.

    Google Scholar 

  53. Neves, C., and Fraga Alves, M. I. (2008). The ratio of maximum to the sum for testing super heavy tails. In Advances in Mathematical and Statistical Modeling (B. Arnold, N. Balakrishnan, J. Sarabia and R. Mínguez eds). Birkhäuser, Boston.

    Google Scholar 

  54. Pancheva, E.I. (1985). Limit theorems for extreme order statistics under nonlinear normalization. In Stability Problems for Stochastic Models, (V.V. Kalashnikov and V.M. Zolotarev eds.), Lecture Notes in Math. 1155, 248- 309, Springer, Berlin.

    Google Scholar 

  55. Pickands III., J. (1975). Statistical inference using extreme order statistics. Ann. Statist. 3, 119-131.

    Article  MATH  MathSciNet  Google Scholar 

  56. Reiss, R.-D. (1981). Uniform approximation to distributions of extreme order statistics. Adv. Appl. Probab. 13, 533-547.

    Article  MATH  MathSciNet  Google Scholar 

  57. Reiss, R.-D. (1987). Estimating the tail index of the claim size distribution. Blätter der DGVM, Band XVIII, 21-25.

    Article  Google Scholar 

  58. Reiss, R.-D. (1989). Approximate Distributions of Order Statistics. (With Applications to Nonparametric Statistics). Springer Series in Statistics, Springer, New York.

    Google Scholar 

  59. Reiss, R.-D., and Thomas, M. (1997). Statistical Analysis of Extreme Values 1st ed., Birkhäuser, Basel.

    MATH  Google Scholar 

  60. Reiss, R.-D. and Thomas, M. (2001). Statistical Analysis of Extreme Values 2nd ed., Birkhäuser, Basel.

    MATH  Google Scholar 

  61. Resnick S.I. (1986). Point processes, regular variation and weak convergences. Adv. Appl. Probab. 18, 66-138.

    Article  MATH  MathSciNet  Google Scholar 

  62. Smith, R.L. (1989). Extreme value analysis of environmental time series: an application to trend detection in ground-level ozone (with discussion). Statistical Science 4, 367-393.

    Article  MATH  MathSciNet  Google Scholar 

  63. Subramanya, U.R. (1994). On max domains of attraction of univariate p-max stable laws. Statist. Probab. Letters 19, 271-279.

    Article  MATH  MathSciNet  Google Scholar 

  64. Taleb, N.N. (2007). The Black Swan: the Impact of the Highly Improbable. Random House.

    Google Scholar 

  65. Teugels, J.L. (1981). Remarks on large claims. Bull. Inst. Internat. Statist. 49, 1490-1500.

    MATH  MathSciNet  Google Scholar 

  66. Teugels, J.L. (1984). Extreme values in insurance mathematics. In Statistical Extremes and Applications. NATO ASI Series, Reidel, Dordrecht.

    Google Scholar 

  67. Villaseñor-Alva, J.A., González-Estrada, E., and Reiss, R.-D. (2006). A nonparametric goodness of fit test for the generalized Pareto distribution. Unpublished manuscript.

    Google Scholar 

  68. Weinstein, S.B. (1973). Theory and applications of some classical and generalized asymptotic distributions of extreme values. IEEE Trans. Inf. Theory 19, 148-154.

    Article  MATH  Google Scholar 

  69. Weiss, L. (1971). Asymptotic inference about a density function at an end of its range. Nav. Res. Logist. Quart. 18, 111-114.

    Article  MATH  Google Scholar 

  70. Weissman, I. (1978). Estimation of parameters and large quantiles based on the k largest observations. J. Amer. Statist. Assoc. 73, 812-815.

    Article  MATH  MathSciNet  Google Scholar 

  71. Witting, H. (1985). Mathematische Statistik I. Teubner, Stuttgart.

    MATH  Google Scholar 

  72. Worms, R. (1998). Propriété asymptotique des excès additifs et valeurs extrêmes: le cas de la loi de Gumbel. C. R. Acad. Sci. Paris, t. 327, Série I, 509-514.

    MATH  MathSciNet  Google Scholar 

  73. Zaliapin, I.V., Kagan, Y.Y., and Schoenberg, F.P. (2005). Approximating the Distribution of Pareto Sums. Pure Appl. Geophys., 162, 1187-1228.

    Article  Google Scholar 

  74. Zeevi, A., and Glynn, P.W. (2004). Recurrence properties of autoregressive processes with super-heavy-tailed innovation. J. Appl. Probab. 41, 639-653.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Würzburg, Am Hubland, 97074, Würzburg, Germany

    Michael Falk

  2. Department of Mathematical Statistics and Actuarial Science, University of Berne, Sidlerstrasse 5, 3012, Bern, Switzerland

    Jürg Hüsler

  3. Department of Mathematics, University of Siegen, Walter Flex Str. 3, 57068, Siegen, Germany

    Rolf-Dieter Reiss

Authors
  1. Michael Falk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jürg Hüsler
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rolf-Dieter Reiss
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Falk .

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Basel AG

About this chapter

Cite this chapter

Falk, M., Hüsler, J., Reiss, RD. (2011). Extreme Value Theory. In: Laws of Small Numbers: Extremes and Rare Events. Springer, Basel. https://doi.org/10.1007/978-3-0348-0009-9_2

Download citation

Publish with us

Policies and ethics

Search

Navigation