IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v98y2007i9p1751-1764.html
   My bibliography  Save this article

Entropy estimate for high-dimensional monotonic functions

Author

Listed:
  • Gao, Fuchang
  • Wellner, Jon A.

Abstract

We establish upper and lower bounds for the metric entropy and bracketing entropy of the class of d-dimensional bounded monotonic functions under Lp norms. It is interesting to see that both the metric entropy and bracketing entropy have different behaviors for p d/(d-1). We apply the new bounds for bracketing entropy to establish a global rate of convergence of the MLE of a d-dimensional monotone density.

Suggested Citation

  • Gao, Fuchang & Wellner, Jon A., 2007. "Entropy estimate for high-dimensional monotonic functions," Journal of Multivariate Analysis, Elsevier, vol. 98(9), pages 1751-1764, October.
    • Handle: RePEc:eee:jmvana:v:98:y:2007:i:9:p:1751-1764
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047-259X(06)00141-2
    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. Polonik, W., 1995. "Density Estimation under Qualitative Assumptions in Higher Dimensions," Journal of Multivariate Analysis, Elsevier, vol. 55(1), pages 61-81, October.
    2. Biau, Gérard & Devroye, Luc, 2003. "On the risk of estimates for block decreasing densities," Journal of Multivariate Analysis, Elsevier, vol. 86(1), pages 143-165, July.
    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. Toru Kitagawa & Shosei Sakaguchi & Aleksey Tetenov, 2021. "Constrained Classification and Policy Learning," Papers 2106.12886, arXiv.org, revised Jul 2023.

    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. Pavlides, Marios G. & Wellner, Jon A., 2012. "Nonparametric estimation of multivariate scale mixtures of uniform densities," Journal of Multivariate Analysis, Elsevier, vol. 107(C), pages 71-89.
    2. Chacón, José E. & Fernández Serrano, Javier, 2024. "Bayesian taut splines for estimating the number of modes," Computational Statistics & Data Analysis, Elsevier, vol. 196(C).
    3. Polonik, Wolfgang, 1997. "Minimum volume sets and generalized quantile processes," Stochastic Processes and their Applications, Elsevier, vol. 69(1), pages 1-24, July.
    4. Nguyen, Hung T. & Wu, Berlin, 2006. "Random and fuzzy sets in coarse data analysis," Computational Statistics & Data Analysis, Elsevier, vol. 51(1), pages 70-85, November.
    5. Neumeyer, Natalie, 2005. "A note on uniform consistency of monotone function estimators," Technical Reports 2005,35, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    6. Obereder, Andreas & Scherzer, Otmar & Kovac, Arne, 2007. "Bivariate density estimation using BV regularisation," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 5622-5634, August.
    7. Polonik, Wolfgang & Yao, Qiwei, 2002. "Set-Indexed Conditional Empirical and Quantile Processes Based on Dependent Data," Journal of Multivariate Analysis, Elsevier, vol. 80(2), pages 234-255, February.
    8. Berlinet, Alain & Biau, Gérard & Rouvière, Laurent, 2005. "Optimal L1 bandwidth selection for variable kernel density estimates," Statistics & Probability Letters, Elsevier, vol. 74(2), pages 116-128, September.
    9. Neumeyer, Natalie, 2007. "A note on uniform consistency of monotone function estimators," Statistics & Probability Letters, Elsevier, vol. 77(7), pages 693-703, April.
    10. Biau, Gérard & Devroye, Luc, 2003. "On the risk of estimates for block decreasing densities," Journal of Multivariate Analysis, Elsevier, vol. 86(1), pages 143-165, July.
    11. De-Jun Feng & Ding Feng, 2004. "On a Statistical Framework for Estimation from Random Set Observations," Journal of Theoretical Probability, Springer, vol. 17(1), pages 85-110, January.

    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:jmvana:v:98:y:2007:i:9:p:1751-1764. 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.