Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Randomly Stopped \({\varvec{k}}\)th Order Statistics

  • Conference paper
  • First Online:
Theory and Practice of Risk Assessment

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 136))

  • 1129 Accesses

Abstract

Randomly stopped order statistics when the stopping rule is generated by a basic count distribution are investigated. Unified expressions in terms of the subordinator are presented, extending results from geometrically thinned sequences. Using the results on limit stable distributions for max-geometric laws, and Smirnov’s techniques to deal with limit laws of extreme order statistics, some results on stability of Panjer subordinated randomly stopped order statistics are discussed.

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
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

Institutional subscriptions

References

  1. Engen, S.: On species frequency models. Biometrika 61, 263–270 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  2. Galambos, J.: The Asymptotic Theory of Extreme Order Statistics. Wiley, New York (1987)

    MATH  Google Scholar 

  3. Hess, K.T., Liewald, A., Schmidt, K.D.: An extension of Panjer’s recursion. ASTIN Bull. 32, 283–297 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Katz, L.: Unified treatment of a broad class of discrete probability distributions. In: Patil, G.P. (ed.) Classical and Contagious Discrete Distributions, pp. 175–182. Pergamon Press, Oxford (1965)

    Google Scholar 

  5. Morris, C.L.: Natural exponential families with quadratic variance functions. Ann. Stat. 10, 65–80 (1982)

    Article  MATH  Google Scholar 

  6. Panjer, H.H.: Recursive evaluation of a family of compound distributions. ASTIN Bull. 12, 22–26 (1981)

    MathSciNet  Google Scholar 

  7. Pestana, D., Velosa, S.: Extensions of Katz-Panjer families of discrete distributions. REVSTAT 2, 145–162 (2004)

    MATH  MathSciNet  Google Scholar 

  8. Rachev, S.T., Resnick, S.: Max-geometric infinite divisibility and stability. Commun. Stat. —Stoch. Models 7, 191–218 (1991)

    Google Scholar 

  9. Rólski, T., Schmidli, H., Schmidt, V., Teugels, J.: Stochastic Processes for Insurance and Finance. Wiley, New York (1999)

    Book  MATH  Google Scholar 

  10. Smirnov, N. V.: Limit distributions for the terms of a variational series. Trudy Mat. Inst. Steklov., Acad. Sci. USSR, Moscow-Leningrad 25, 3–60 (1949)

    Google Scholar 

  11. Sundt, B.: On some extensions of Panjer’s class of counting distributions. ASTIN Bull. 22, 61–80 (1992)

    Article  Google Scholar 

  12. Sundt, B., Jewell, W.: Further results on recursive evaluation of compound distributions. ASTIN Bull. 12, 27–39 (1981)

    MathSciNet  Google Scholar 

  13. Willmot, G.E.: Sundt and Jewell’s family of discrete distributions. ASTIN Bull. 18, 17–29 (1988)

    Article  Google Scholar 

Download references

Acknowledgments

The authors are grateful to the referees for many valuable suggestions that have been used to improve the presentation of the paper. Professor Sneh Gulati’s very thorough proofreading of the text is gratefully acknowledged.

This research has been supported by National Funds through FCT – Fundação para a Ciência e a Tecnologia, projects PEst-OE/MAT/UI0006/2011, PEst-OE/MAT/UI0006/2014, and EXTREMA, PTDC /MAT /101736/2008.

Author information

Authors and Affiliations

  1. CCEE, Universidade da Madeira, 9000-390, Funchal, Portugal

    Sandra Mendonça

  2. CEAUL, Universidade de Lisboa, Lisbon, Portugal

    Sandra Mendonça

  3. CEAUL and DEIO–FCUL, Universidade de Lisboa, Campo Grande, 1749-016, Lisbon, Portugal

    Dinis Pestana & M. Ivette Gomes

  4. Instituto de Investigação Científica Bento da Rocha Cabral, Lisbon, Portugal

    Dinis Pestana & M. Ivette Gomes

Authors
  1. Sandra Mendonça
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dinis Pestana
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Ivette Gomes
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sandra Mendonça .

Editor information

Editors and Affiliations

  1. Technological Educational Institute of Athens, Athens, Greece

    Christos P. Kitsos

  2. Universidade Aberta, Palácio Ceia, Lisbon, Portugal

    Teresa A. Oliveira

  3. Democritus University, Thrace, Germany

    Alexandros Rigas

  4. Florida International University, Miami, Florida, USA

    Sneh Gulati

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Mendonça, S., Pestana, D., Gomes, M.I. (2015). Randomly Stopped \({\varvec{k}}\)th Order Statistics. In: Kitsos, C., Oliveira, T., Rigas, A., Gulati, S. (eds) Theory and Practice of Risk Assessment. Springer Proceedings in Mathematics & Statistics, vol 136. Springer, Cham. https://doi.org/10.1007/978-3-319-18029-8_19

Download citation

Publish with us

Policies and ethics

Search

Navigation