IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v80y2010i13-14p1093-1102.html
   My bibliography  Save this article

The distribution and quantiles of functionals of weighted empirical distributions when observations have different distributions

Author

Listed:
  • Withers, Christopher S.
  • Nadarajah, Saralees

Abstract

This paper extends Edgeworth-Cornish-Fisher expansions for the distribution and quantiles of nonparametric estimates in two ways. Firstly, it allows observations to have different distributions. Secondly, it allows the observations to be weighted in a predetermined way. The use of weighted estimates has a long history, including applications to regression, rank statistics and Bayes theory. However, asymptotic results have generally been only first order (the CLT and weak convergence). We give third order asymptotics for the distribution and percentiles of any smooth functional of a weighted empirical distribution, thus allowing a considerable increase in accuracy over earlier CLT results. Consider independent non-identically distributed (non-iid) observations X1n,...,Xnn in Rs. Let be their weighted empirical distribution with weights w1n,...,wnn. We obtain cumulant expansions and hence Edgeworth-Cornish-Fisher expansions for for any smooth functional T([dot operator]) by extending the concepts of von Mises derivatives to signed measures of total measure 1. As an example we give the cumulant coefficients needed for Edgeworth-Cornish-Fisher expansions to O(n-3/2) for the sample coefficient of variation when observations are non-iid.

Suggested Citation

  • Withers, Christopher S. & Nadarajah, Saralees, 2010. "The distribution and quantiles of functionals of weighted empirical distributions when observations have different distributions," Statistics & Probability Letters, Elsevier, vol. 80(13-14), pages 1093-1102, July.
    • Handle: RePEc:eee:stapro:v:80:y:2010:i:13-14:p:1093-1102
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167-7152(10)00073-8
    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. Christopher Withers & Saralees Nadarajah, 2008. "Edgeworth expansions for functions of weighted empirical distributions with applications to nonparametric confidence intervals," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(8), pages 751-768.
    2. Lahiri, S. N., 1992. "Edgeworth expansions for M-estimators of a regression parameter," Journal of Multivariate Analysis, Elsevier, vol. 43(1), pages 125-132, October.
    3. C. Withers, 1988. "Nonparametric confidence intervals for functions of several distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 40(4), pages 727-746, December.
    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. Joanna Janczura & Rafał Weron, 2013. "Goodness-of-fit testing for the marginal distribution of regime-switching models with an application to electricity spot prices," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 97(3), pages 239-270, July.
    2. Janczura, Joanna & Weron, Rafal, 2010. "Goodness-of-fit testing for regime-switching models," MPRA Paper 22871, University Library of Munich, Germany.

    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. Christopher S. Withers & Saralees Nadarajah, 2022. "Cornish-Fisher Expansions for Functionals of the Weighted Partial Sum Empirical Distribution," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1791-1804, September.
    2. Christopher Withers & Saralees Nadarajah, 2008. "Edgeworth expansions for functions of weighted empirical distributions with applications to nonparametric confidence intervals," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(8), pages 751-768.
    3. Christopher Withers & Saralees Nadarajah, 2013. "Correlation is first order independent of transformation," Statistical Papers, Springer, vol. 54(2), pages 443-456, May.
    4. Withers, Christopher S. & Nadarajah, Saralees, 2009. "Accurate tests and intervals based on linear cusum statistics," Statistics & Probability Letters, Elsevier, vol. 79(5), pages 689-697, March.
    5. Christopher S. Withers & Saralees Nadarajah, 2014. "Expansions about the Gamma for the Distribution and Quantiles of a Standard Estimate," Methodology and Computing in Applied Probability, Springer, vol. 16(3), pages 693-713, September.
    6. Christopher Withers & Saralees Nadarajah, 2010. "Expansions for log densities of asymptotically normal estimates," Statistical Papers, Springer, vol. 51(2), pages 247-257, June.
    7. Withers, Christopher S. & Nadarajah, Saralees, 2012. "Improved confidence regions based on Edgeworth expansions," Computational Statistics & Data Analysis, Elsevier, vol. 56(12), pages 4366-4380.
    8. Withers, Christopher S. & Nadarajah, Saralees, 2009. "Accurate tests and intervals based on nonlinear cusum statistics," Statistics & Probability Letters, Elsevier, vol. 79(21), pages 2242-2250, November.
    9. Christopher Withers & Saralees Nadarajah, 2010. "Tilted Edgeworth expansions for asymptotically normal vectors," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(6), pages 1113-1142, December.
    10. Christopher Withers & Saralees Nadarajah, 2012. "On the dependency for asymptotically independent estimates," Statistical Inference for Stochastic Processes, Springer, vol. 15(2), pages 127-132, July.
    11. Christopher Withers & Saralees Nadarajah, 2012. "Unbiased estimates for a lognormal regression problem and a nonparametric alternative," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 75(2), pages 207-227, February.
    12. Withers, Christopher S. & Nadarajah, Saralees, 2019. "Moments and cumulants of the extremes of a sample from a uniform distribution," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 238-247.

    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:80:y:2010:i:13-14:p:1093-1102. 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.