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Entropy estimate for high-dimensional monotonic functions


  • Gao, Fuchang
  • Wellner, Jon A.


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

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    References listed on IDEAS

    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, GĂ©rard & Devroye, Luc, 2003. "On the risk of estimates for block decreasing densities," Journal of Multivariate Analysis, Elsevier, vol. 86(1), pages 143-165, July.
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