Skip to main content
SpringerLink
Log in

On Discrimination between Classes of Distribution Tails

  • Methods of Signal Processing
  • Published:
Problems of Information Transmission Aims and scope Submit manuscript

Abstract

We propose a test to distinguish between two classes of distribution tails using only higher order statistics of a sample and prove its consistency. We do not assume the corresponding distribution functions to belong to any maximum domain of attraction.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Goldie, C.M. and Smith, R.L., Slow Variation with Remainder: Theory and Applications, Quart. J. Math. Oxford Ser. (2), 1987, vol. 38, no. 1, pp. 45–71.

    Article  MathSciNet  MATH  Google Scholar 

  2. Guilbaud, O., Exact Kolmogorov-type Test for Left-Truncated and/or Right-Censored Data, J. Amer. Statist. Assoc., 1988, vol. 83, no. 401, pp. 213–221.

    Article  MathSciNet  MATH  Google Scholar 

  3. Chernobai, A., Menn, C., Rachev, S.T., and Trück, S., Estimation of Operational Value-at-Risk in the Presence of Minimum Collection Thresholds, Tech. Report of the Univ. of California, Santa Barbara, CA, USA, 2005.

    Google Scholar 

  4. Beirlant, T., Goegebeur, Y., Teugels, J., and Segers, J., Statistics of Extremes: Theory and Applications, Hoboken, NJ: Wiley, 2004.

    Book  MATH  Google Scholar 

  5. de Haan, L. and Ferreira, A., Extreme Value Theory: An Introduction, New York: Springer, 2006.

    Book  MATH  Google Scholar 

  6. Gnedenko, B.V., Sur la distribution limite du terme maximum d’une série aléatoire, Ann. Math. Ser. (2), 1943, vol. 44, no. 3, pp. 423–453.

    Article  MathSciNet  MATH  Google Scholar 

  7. Bingham, N.H., Goldie, C.M., and Teugels, J.L., Regular Variation, Cambridge: Cambridge Univ. Press, 1987.

    Book  MATH  Google Scholar 

  8. de Haan, L. and Resnick, S., Second-Order Regular Variation and Rates of Convergence in Extreme-Value Theory, Ann. Probab., 1996, vol. 24, no. 1, pp. 97–124.

    Article  MathSciNet  MATH  Google Scholar 

  9. Pickands, J., III, Statistical Inference Using Extreme Order Statistics, Ann. Statist., 1975, vol. 3, no. 1, pp. 119–131.

    MathSciNet  Google Scholar 

  10. Hall, P., On Some Simple Estimates of an Exponent of Regular Variation, J. Roy. Statist. Soc. Ser. B, 1982, vol. 44, no. 1, pp. 37–42.

    MathSciNet  MATH  Google Scholar 

  11. Beirlant, T. and Teugels, J.L., Asymptotics of Hill’s Estimator, Teor. Veroyatnost. i Primenen., 1986, vol. 31, no. 3, pp. 530–536 [Theory Probab. Appl. (Reprint), 1987, vol. 31, no. 3, pp. 463–469].

    MathSciNet  MATH  Google Scholar 

  12. Martins, M.J., Estimaçao de Caudas Pesadas–Variantes ao Estimador de Hill (Heavy Tails Estimation— Variants to the Hill Estimator), PhD Thesis, Univ. of Lisbon, Portugal, 2000.

    Google Scholar 

  13. Fraga Alves, M.I., de Haan, L., and Lin, T., Estimation of the Parameter Controlling the Speed of Convergence in Extreme Value Theory, Math. Methods Statist., 2003, vol. 12, no. 2, pp. 155–176.

    MathSciNet  Google Scholar 

  14. Drees, H., Ferreira, A., and de Haan, L., On Maximum Likelihood Estimation of the Extreme Value Index, Ann. Appl. Probab., 2004, vol. 14, no. 3, pp. 1179–1201.

    Article  MathSciNet  MATH  Google Scholar 

  15. Fraga Alves, M.I., Gomes, M.I., and de Haan, L., A New Class of Semi-Parametric Estimators of the Second Order Parameter, Port. Math. (N.S.), 2003, vol. 60, no. 2, pp. 193–213.

    MathSciNet  Google Scholar 

  16. Smith, R.L., Estimating Tails of Probability Distributions, Ann. Statist., 1987, vol. 15, no. 3, pp. 1174–1207.

    Article  MathSciNet  MATH  Google Scholar 

  17. de Haan, L. and Sinha, A.K., Estimating the Probability of a Rare Event, Ann. Statist., 1999, vol. 27, no. 2, pp. 732–759.

    Article  MathSciNet  MATH  Google Scholar 

  18. Drees, H., de Haan, L., and Li, D., Approximations to the Tail Empirical Distribution Function with Application to Testing Extreme Value Conditions, J. Statist. Plann. Inference, 2006, vol. 136, no. 10, pp. 3498–3538.

    Article  MathSciNet  MATH  Google Scholar 

  19. Fraga Alves, I., de Haan, L., and Neves, C., A Test Procedure for Detecting Super-Heavy Tails, J. Statist. Plann. Inference, 2009, vol. 139, no. 2, pp. 213–227.

    Article  MathSciNet  MATH  Google Scholar 

  20. Gardes, L., Girard, S., and Guillou, A., Weibull Tail-Distributions Revisited: A New Look at Some Tail Estimators, J. Statist. Plann. Inference, 2009, vol. 141, no. 1, pp. 429–444.

    Article  MathSciNet  MATH  Google Scholar 

  21. Hill, B.M., A Simple General Approach to Inference about the Tail of a Distribution, Ann. Statist., 1975, vol. 3, no. 5, pp. 1163–1174.

    Article  MathSciNet  MATH  Google Scholar 

  22. Berred, M., Record Values and the Estimation of the Weibull Tail-Coefficient, C. R. Acad. Sci. Paris Sér. I Math., 1991, vol. 312, no. 12, pp. 943–946.

    MathSciNet  MATH  Google Scholar 

  23. Diebolt, J., Gardes, L., Girard, S., and Guillou, A., Bias-Reduced Estimators of the Weibull Tail-Coefficient, TEST, 2008, vol. 17, no. 2, pp. 311–331.

    Article  MathSciNet  MATH  Google Scholar 

  24. Rodionov, I.V., Discrimination of Close Hypotheses about Distribution Tails Using Higher Order Statistics, to appear in Teor. Veroyatnost. i Primenen.

  25. Rodionov, I.V., A Discrimination Test for Tails of Weibull-type Distributions, Teor. Veroyatnost. i Primenen., 2018, vol. 63, no. 2, pp. 402–413.

    Article  Google Scholar 

  26. Gardes, L. and Girard, S., Comparison of Weibull Tail-Coefficient Estimators, REVSTAT, 2006, vol. 4, no. 2, pp. 163–188.

    MathSciNet  MATH  Google Scholar 

  27. Fraga Alves, M.I. and Neves, C., Reiss and Thomas’ Automatic Selection of the Number of Extremes, Comput. Statist. Data Anal., 2004, vol. 47, no. 4, pp. 689–704.

    Article  MathSciNet  MATH  Google Scholar 

  28. Falk, M., Some Best Estimators for Distributions with Finite Endpoint, Statistics, 1995, vol. 27, no. 1–2, pp. 115–125.

    Article  MathSciNet  MATH  Google Scholar 

  29. Resnick, S.I., A Probability Path, Boston: Birkhäuser, 1999.

    MATH  Google Scholar 

  30. Mikusheva, A.E., The Law of Large Numbers and the Logarithmic Law for Arrays, Fundam. Prikl. Mat., 2000, vol. 6, no. 1, pp. 195–206.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

    I. V. Rodionov

  2. Department of Innovation and High Technology, Moscow Institute of Physics and Technology (State University), Moscow, Russia

    I. V. Rodionov

Authors
  1. I. V. Rodionov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. V. Rodionov.

Additional information

Original Russian Text © I.V. Rodionov, 2018, published in Problemy Peredachi Informatsii, 2018, Vol. 54, No. 2, pp. 29–44.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rodionov, I.V. On Discrimination between Classes of Distribution Tails. Probl Inf Transm 54, 124–138 (2018). https://doi.org/10.1134/S0032946018020035

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0032946018020035

Search

Navigation