IDEAS home Printed from https://ideas.repec.org/a/spr/stpapr/v58y2017i4d10.1007_s00362-015-0734-7.html
   My bibliography  Save this article

Application of the delta method to functions of the sample mean when observations are dependent

Author

Listed:
  • Michael Weba

    (Goethe University Frankfurt)

  • Nora Dörmann

    (Goethe University Frankfurt)

Abstract

Let $$\mu $$ μ be the expected value of a random variable and $$\bar{X}_n$$ X ¯ n the corresponding sample mean of n observations. If the transformed expectation $$f(\mu )$$ f ( μ ) is to be estimated by $$f\left( \bar{X}_n\right) $$ f X ¯ n then the delta method is a widely used tool to describe the asymptotic behaviour of $$f\left( \bar{X}_n\right) $$ f X ¯ n . Regarding bias and variance, however, conventional theorems require independent observations as well as boundedness conditions of f being violated even by “simple” functions such as roots or logarithms. It is shown that asymptotic expansions for bias and variance still hold if restrictive boundedness conditions are replaced by considerably weaker requirements upon the global growth of f. Moreover, observations are allowed to be dependent.

Suggested Citation

  • Michael Weba & Nora Dörmann, 2017. "Application of the delta method to functions of the sample mean when observations are dependent," Statistical Papers, Springer, vol. 58(4), pages 957-986, December.
    • Handle: RePEc:spr:stpapr:v:58:y:2017:i:4:d:10.1007_s00362-015-0734-7
    DOI: 10.1007/s00362-015-0734-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00362-015-0734-7
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s00362-015-0734-7?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    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. Afshin Parvardeh, 2015. "A note on the asymptotic distribution of the estimation of the mean past lifetime," Statistical Papers, Springer, vol. 56(1), pages 205-215, February.
    2. Beutner, Eric & Zähle, Henryk, 2010. "A modified functional delta method and its application to the estimation of risk functionals," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2452-2463, 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. Eric Beutner & Henryk Zähle, 2018. "Bootstrapping Average Value at Risk of Single and Collective Risks," Risks, MDPI, vol. 6(3), pages 1-30, September.
    2. Mario Ghossoub & Jesse Hall & David Saunders, 2020. "Maximum Spectral Measures of Risk with given Risk Factor Marginal Distributions," Papers 2010.14673, arXiv.org.
    3. Tino Werner, 2022. "Asymptotic linear expansion of regularized M-estimators," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(1), pages 167-194, February.
    4. Daniel Bartl & Ludovic Tangpi, 2020. "Non-asymptotic convergence rates for the plug-in estimation of risk measures," Papers 2003.10479, arXiv.org, revised Oct 2022.
    5. Krätschmer, Volker & Schied, Alexander & Zähle, Henryk, 2012. "Qualitative and infinitesimal robustness of tail-dependent statistical functionals," Journal of Multivariate Analysis, Elsevier, vol. 103(1), pages 35-47, January.
    6. Jafar Ahmadi & H. N. Nagaraja, 2020. "Conditional properties of a random sample given an order statistic," Statistical Papers, Springer, vol. 61(5), pages 1971-1996, October.
    7. Volker Krätschmer & Henryk Zähle, 2017. "Statistical Inference for Expectile-based Risk Measures," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(2), pages 425-454, June.
    8. Krätschmer Volker & Schied Alexander & Zähle Henryk, 2015. "Quasi-Hadamard differentiability of general risk functionals and its application," Statistics & Risk Modeling, De Gruyter, vol. 32(1), pages 25-47, April.
    9. Buchsteiner, Jannis, 2015. "Weak convergence of the weighted sequential empirical process of some long-range dependent data," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 170-179.
    10. Soren Bettels & Stefan Weber, 2024. "An Integrated Approach to Importance Sampling and Machine Learning for Efficient Monte Carlo Estimation of Distortion Risk Measures in Black Box Models," Papers 2408.02401, arXiv.org.
    11. Beare, Brendan K. & Shi, Xiaoxia, 2019. "An improved bootstrap test of density ratio ordering," Econometrics and Statistics, Elsevier, vol. 10(C), pages 9-26.
    12. Manon Costa & Sébastien Gadat, 2021. "Non-asymptotic study of a recursive superquantile estimation algorithm," Post-Print hal-03610477, HAL.
    13. Labopin-Richard T. & Gamboa F. & Garivier A. & Iooss B., 2016. "Bregman superquantiles. Estimation methods and applications," Dependence Modeling, De Gruyter, vol. 4(1), pages 1-33, March.
    14. Krätschmer, Volker & Zähle, Henryk, 2011. "Sensitivity of risk measures with respect to the normal approximation of total claim distributions," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 335-344.
    15. Volker Krätschmer & Alexander Schied & Henryk Zähle, 2014. "Comparative and qualitative robustness for law-invariant risk measures," Finance and Stochastics, Springer, vol. 18(2), pages 271-295, April.
    16. Darinka Dentcheva & Spiridon Penev & Andrzej Ruszczyński, 2017. "Statistical estimation of composite risk functionals and risk optimization problems," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(4), pages 737-760, August.
    17. Gadat, Sébastien & Costa, Manon, 2020. "Non asymptotic controls on a stochastic algorithm for superquantile approximation," TSE Working Papers 20-1149, Toulouse School of Economics (TSE).
    18. Elham Zamanzade & Majid Asadi & Afshin Parvardeh & Ehsan Zamanzade, 2023. "A ranked-based estimator of the mean past lifetime with an application," Statistical Papers, Springer, vol. 64(1), pages 161-177, February.
    19. Arnold Janssen & Andreas Knoch, 2016. "Information bounds for nonparametric estimators of L-functionals and survival functionals under censored data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 79(2), pages 195-220, February.
    20. 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.

    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:spr:stpapr:v:58:y:2017:i:4:d:10.1007_s00362-015-0734-7. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.