IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v79y2009i16p1774-1783.html
   My bibliography  Save this article

Bahadur representations for the median absolute deviation and its modifications

Author

Listed:
  • Mazumder, Satyaki
  • Serfling, Robert

Abstract

The median absolute deviation about the median (MAD) is an important univariate spread measure having wide appeal due to its highly robust sample version. A powerful tool in treating the asymptotics of a statistic is a linearization, i.e., a Bahadur representation. Here we establish both strong and weak Bahadur representations for the sample MAD. The strong version is the first such result in the literature, while the weak version improves upon previous treatments by reducing regularity conditions. Our results also apply to a modified version of sample MAD [Tyler, D.E., 1994. Finite sample breakdown points of projection based multivariate location and scatter statistics. Annals of Statistics 22, 1024-1044; Gather, U., Hilker, T., 1997. A note on Tyler's modification of the MAD for the Stahel-Donoho estimator. Annals of Statistics 25, 2024-2026] introduced to obtain improved robustness for statistical procedures using sample median and MAD combinations over the univariate projections of multivariate data. The strong version yields the law of iterated logarithm for the sample MAD and supports the study of almost sure properties of randomly trimmed means based on the MAD and development of robust sequential nonparametric confidence intervals for the MAD. The weak version is needed to simplify derivations of the asymptotic joint distributions of vectors of dependent sample median and sample MAD combinations, which arise in constructing nonparametric multivariate outlyingness functions via projection pursuit.

Suggested Citation

  • Mazumder, Satyaki & Serfling, Robert, 2009. "Bahadur representations for the median absolute deviation and its modifications," Statistics & Probability Letters, Elsevier, vol. 79(16), pages 1774-1783, August.
    • Handle: RePEc:eee:stapro:v:79:y:2009:i:16:p:1774-1783
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167-7152(09)00186-2
    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. Olive, David J., 2001. "High breakdown analogs of the trimmed mean," Statistics & Probability Letters, Elsevier, vol. 51(1), pages 87-92, January.
    2. Serfling, Robert & Mazumder, Satyaki, 2009. "Exponential probability inequality and convergence results for the median absolute deviation and its modifications," Statistics & Probability Letters, Elsevier, vol. 79(16), pages 1767-1773, August.
    3. Zhiqiang Chen & Evarist Giné, 2004. "Another approach to asymptotics and bootstrap of randomly trimmed means," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 56(4), pages 771-790, December.
    4. Falk, Michael, 1997. "Asymptotic independence of median and MAD," Statistics & Probability Letters, Elsevier, vol. 34(4), pages 341-345, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Marcel, Bräutigam & Marie, Kratz, 2018. "On the Dependence between Quantiles and Dispersion Estimators," ESSEC Working Papers WP1807, ESSEC Research Center, ESSEC Business School.
    2. Marcel Bräutigam & Marie Kratz, 2018. "On the Dependence between Quantiles and Dispersion Estimators," Working Papers hal-02296832, HAL.
    3. Nagatsuka, Hideki & Kawakami, Hiroshi & Kamakura, Toshinari & Yamamoto, Hisashi, 2013. "The exact finite-sample distribution of the median absolute deviation about the median of continuous random variables," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 999-1005.

    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. Serfling, Robert & Mazumder, Satyaki, 2009. "Exponential probability inequality and convergence results for the median absolute deviation and its modifications," Statistics & Probability Letters, Elsevier, vol. 79(16), pages 1767-1773, August.
    2. Nagatsuka, Hideki & Kawakami, Hiroshi & Kamakura, Toshinari & Yamamoto, Hisashi, 2013. "The exact finite-sample distribution of the median absolute deviation about the median of continuous random variables," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 999-1005.
    3. Olive, David J., 2001. "High breakdown analogs of the trimmed mean," Statistics & Probability Letters, Elsevier, vol. 51(1), pages 87-92, January.
    4. Marcel, Bräutigam & Marie, Kratz, 2018. "On the Dependence between Quantiles and Dispersion Estimators," ESSEC Working Papers WP1807, ESSEC Research Center, ESSEC Business School.
    5. Zhiqiang Chen & Evarist Giné, 2004. "Another approach to asymptotics and bootstrap of randomly trimmed means," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 56(4), pages 771-790, December.
    6. Mahesh K.C & Arnab Kumar Laha, 2021. "A Robust Sharpe Ratio," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(2), pages 444-465, November.
    7. Wang, Shanshan & Serfling, Robert, 2018. "On masking and swamping robustness of leading nonparametric outlier identifiers for multivariate data," Journal of Multivariate Analysis, Elsevier, vol. 166(C), pages 32-49.
    8. Marcel Bräutigam & Marie Kratz, 2018. "On the Dependence between Quantiles and Dispersion Estimators," Working Papers hal-02296832, HAL.

    More about this item

    Statistics

    Access and download statistics

    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:stapro:v:79:y:2009:i:16:p:1774-1783. 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/wps/find/journaldescription.cws_home/622892/description#description .

    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.