Skip to main content
SpringerLink
Log in
Book cover

Stochastic Orders in Reliability and Risk pp 313–330Cite as

Second-Order Conditions of Regular Variation and Drees-Type Inequalities

  • Chapter
  • First Online:

Part of the book series: Lecture Notes in Statistics ((LNSP,volume 208))

Abstract

There are a variety of concepts extending regular variation, among which are the extended regular variation (ERV), second-order regular variation (2RV), and second-order extended regular variation (2ERV). In this paper, we reexamine the connections from 2ERV to 2RV and recover and strengthen the main result in Neves (2009, [35]) by using a different but straightforward approach. We also present new Drees-type inequalities in which the original auxiliary functions of a 2ERV or ERV function are not replaced by other ones.

This is a preview of subscription content, 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   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover 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

Learn about institutional subscriptions

Bibliography

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

    MATH  Google Scholar 

  2. Cheng, S. and Jiang, C.: The Edgeworth expansion for distributions of extreme values. Science in China, 44, 427–437 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. de Haan, L. and Ferreira, A.: Extreme Value Theory: An introduction. Springer, New York (2006)

    MATH  Google Scholar 

  4. de Haan, L. and Stadtmüller, U.: Generalized regular variation of second order. Journal of the Australian Mathematical Society, 61, 381–395 (1996)

    Article  MATH  Google Scholar 

  5. Drees, H.: On smooth statistical tail functionals. Scandinavian Journal of Statistics, 25, 187–210 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fraga Alves, M. I., Gomes, M. I., de Haan, L. and Neves, C.: A note on second order conditions in extreme value theory: Linking general and heavy tails conditions. Revstat, 5, 285–304 (2007)

    MathSciNet  MATH  Google Scholar 

  7. Hua, L. and Joe, H.: Second order regular variation and conditional tail expectation of multiple risks. Insurance: Mathematics and Economics 49, 537–546 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lv, W., Mao, T. and Hu, T.: Properties of second-order regular variation and expansions for risk concentration. Probability in the Engineering and Informational Sciences, To appear (2012)

    Google Scholar 

  9. Neves, C.: From extended regular variation to regular variation with application in extreme value statistics. Journal of Mathematical Analysis and Applications, 355, 216–230 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Resnick, S. I.: Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York (2007)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui, 230026, China

    Tiantian Mao

Authors
  1. Tiantian Mao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tiantian Mao .

Editor information

Editors and Affiliations

  1. Dept. Mathematics, Washington State University, Lighty 370, Pullman, 99164, Washington, USA

    Haijun Li

  2. Xiamen University, Xiamen, 361000, China, People's Republic

    Xiaohu Li

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this chapter

Cite this chapter

Mao, T. (2013). Second-Order Conditions of Regular Variation and Drees-Type Inequalities. In: Li, H., Li, X. (eds) Stochastic Orders in Reliability and Risk. Lecture Notes in Statistics(), vol 208. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6892-9_16

Download citation

Publish with us

Policies and ethics

Search

Navigation