IDEAS home Printed from https://ideas.repec.org/a/eee/insuma/v47y2010i3p352-357.html
   My bibliography  Save this article

Bounds for the bias of the empirical CTE

Author

Listed:
  • Russo, Ralph P.
  • Shyamalkumar, Nariankadu D.

Abstract

The Conditional Tail Expectation (CTE) is gaining an increasing level of attention as a measure of risk. It is known that nonparametric unbiased estimators of the CTE do not exist, and that , the empirical [alpha]-level CTE (the average of the n(1-[alpha]) largest order statistics in a random sample of size n), is negatively biased. In this article, we show that increasing convex order among distributions is preserved by . From this result it is possible to identify the specific distributions, within some large classes of distributions, that maximize the bias of . This in turn leads to best possible bounds on the bias under various sets of conditions on the sampling distribution F. In particular, we show that when the [alpha]-level quantile is an isolated point in the support of a non-degenerate distribution (for example, a lattice distribution) then the bias is either of the order or vanishes exponentially fast. This is intriguing as the bias of vanishes at the in-between rate of 1/n when F possesses a positive derivative at the [alpha]th quantile.

Suggested Citation

  • Russo, Ralph P. & Shyamalkumar, Nariankadu D., 2010. "Bounds for the bias of the empirical CTE," Insurance: Mathematics and Economics, Elsevier, vol. 47(3), pages 352-357, December.
    • Handle: RePEc:eee:insuma:v:47:y:2010:i:3:p:352-357
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167-6687(10)00086-7
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. B. John Manistre & Geoffrey Hancock, 2005. "Variance of the CTE Estimator," North American Actuarial Journal, Taylor & Francis Journals, vol. 9(2), pages 129-156.
    2. Ko, Bangwon & Russo, Ralph P. & Shyamalkumar, Nariankadu D., 2009. "A Note on Nonparametric Estimation of the CTE," ASTIN Bulletin, Cambridge University Press, vol. 39(2), pages 717-734, November.
    3. Bruce Jones & Ričardas Zitikis, 2003. "Empirical Estimation of Risk Measures and Related Quantities," North American Actuarial Journal, Taylor & Francis Journals, vol. 7(4), pages 44-54.
    4. Diane L. Evans & Lawrence M. Leemis & John H. Drew, 2006. "The Distribution of Order Statistics for Discrete Random Variables with Applications to Bootstrapping," INFORMS Journal on Computing, INFORMS, vol. 18(1), pages 19-30, February.
    5. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    6. Kim, Joseph Hyun Tae & Hardy, Mary R., 2007. "Quantifying and Correcting the Bias in Estimated Risk Measures," ASTIN Bulletin, Cambridge University Press, vol. 37(2), pages 365-386, November.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Kim, Joseph H.T. & Jeon, Yongho, 2013. "Credibility theory based on trimming," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 36-47.
    2. Kim, Joseph H.T., 2010. "Bias correction for estimated distortion risk measure using the bootstrap," Insurance: Mathematics and Economics, Elsevier, vol. 47(2), pages 198-205, October.
    3. Gao, Huan & Mamon, Rogemar & Liu, Xiaoming, 2017. "Risk measurement of a guaranteed annuity option under a stochastic modelling framework," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 132(C), pages 100-119.
    4. Ahn, Jae Youn & Shyamalkumar, Nariankadu D., 2014. "Asymptotic theory for the empirical Haezendonck–Goovaerts risk measure," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 78-90.
    5. Alois Pichler, 2013. "Premiums And Reserves, Adjusted By Distortions," Papers 1304.0490, arXiv.org.
    6. Brahimi, Brahim & Meraghni, Djamel & Necir, Abdelhakim & Zitikis, Ričardas, 2011. "Estimating the distortion parameter of the proportional-hazard premium for heavy-tailed losses," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 325-334.
    7. Pichler, Alois & Shapiro, Alexander, 2015. "Minimal representation of insurance prices," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 184-193.
    8. Michael B. Gordy & Sandeep Juneja, 2010. "Nested Simulation in Portfolio Risk Measurement," Management Science, INFORMS, vol. 56(10), pages 1833-1848, October.
    9. Psarrakos, Georgios & Vliora, Polyxeni, 2021. "Sensitivity analysis and tail variability for the Wang’s actuarial index," Insurance: Mathematics and Economics, Elsevier, vol. 98(C), pages 147-152.
    10. Jones, Bruce L. & Puri, Madan L. & Zitikis, Ricardas, 2006. "Testing hypotheses about the equality of several risk measure values with applications in insurance," Insurance: Mathematics and Economics, Elsevier, vol. 38(2), pages 253-270, April.
    11. Bruce L. Jones & Ricardas Zitikis, 2005. "Testing for the order of risk measures: an application of L-statistics in actuarial science," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(2), pages 193-211.
    12. Greselin, Francesca & Zitikis, Ricardas, 2015. "Measuring economic inequality and risk: a unifying approach based on personal gambles, societal preferences and references," MPRA Paper 65892, University Library of Munich, Germany.
    13. Georg Pflug & Nancy Wozabal, 2010. "Asymptotic distribution of law-invariant risk functionals," Finance and Stochastics, Springer, vol. 14(3), pages 397-418, September.
    14. Adam, Alexandre & Houkari, Mohamed & Laurent, Jean-Paul, 2008. "Spectral risk measures and portfolio selection," Journal of Banking & Finance, Elsevier, vol. 32(9), pages 1870-1882, September.
    15. Asimit, Alexandru V. & Li, Jinzhu, 2016. "Extremes for coherent risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 332-341.
    16. Pflug Georg Ch., 2006. "On distortion functionals," Statistics & Risk Modeling, De Gruyter, vol. 24(1/2006), pages 1-16, July.
    17. Francesca Greselin & Ričardas Zitikis, 2018. "From the Classical Gini Index of Income Inequality to a New Zenga-Type Relative Measure of Risk: A Modeller’s Perspective," Econometrics, MDPI, vol. 6(1), pages 1-20, January.
    18. Psarrakos, Georgios & Sordo, Miguel A., 2019. "On a family of risk measures based on proportional hazards models and tail probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 232-240.
    19. Henryk Zähle, 2011. "Rates of almost sure convergence of plug-in estimates for distortion risk measures," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 74(2), pages 267-285, September.
    20. Jones, Bruce L. & Zitikis, Ricardas, 2007. "Risk measures, distortion parameters, and their empirical estimation," Insurance: Mathematics and Economics, Elsevier, vol. 41(2), pages 279-297, September.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:insuma:v:47:y:2010:i:3:p:352-357. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/inca/505554 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.