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Bracketing Entropy of High Dimensional Distributions

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High Dimensional Probability VI

Part of the book series: Progress in Probability ((PRPR,volume 66))

Abstract

Let \(\mathcal{F}_{d}\) be the class of probability distribution functions on \([0,\,1]^{d},\,{d}\geq{2}\). The following estimate for the bracketing entropy of \(\mathcal{F}_{d}\) in the \([L]^{p}\) norm, \(1\,\leq\,p\,{<} \infty \), is obtained:

$${\rm log}{N_{[\,]}}(\varepsilon, \mathcal{F}_{d},{\parallel.\parallel}_p)=O(\varepsilon^{-1}{|\rm log\varepsilon|^{2(\rm d-1)}}).$$

Based on this estimate, a general relation between bracketing entropy in the L p norm and metric entropy in the L 1 norm for multivariate smooth functions is established.

Mathematics Subject Classification (2010). Primary 41A25; Secondary 62G05.

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References

  1. Beck, JĂłzsef and Chen, William W.L. Irregularities of distribution. Cambridge Tracts in Mathematics, 89. Cambridge University Press, Cambridge, 1987.

    Google Scholar 

  2. Bilyk, Dmitriy and Lacey, Michael T. and Vagharshakyan, Armen. On the small ball inequality in all dimensions. J. Funct. Anal. 254 (2008), no. 9, 2470–2502.

    Google Scholar 

  3. Blei, Ron; Gao, Fuchang and Li, Wenbo V. Metric entropy of high dimensional distributions. Proc. Amer. Math. Soc. 135 (2007), no. 12, 4009–4018.

    Google Scholar 

  4. Dunker, T. and Linde, W. and Kühn, T. and Lifshits, M.A. Metric entropy of integration operators and small ball probabilities for the Brownian sheet. J. Approx. Theory 101 (1999), no. 1, 63–77.

    Article  MathSciNet  MATH  Google Scholar 

  5. Gao, Fuchang. Entropy estimate for κ-monotone functions via small ball probability of integrated Brownian motion. Electron. Commun. Probab. 13 (2008), 121–130.

    Google Scholar 

  6. Gao, Fuchang; Li, Wenbo V. and Wellner, Jon A. How many Laplace transforms of probability measures are there? Proc. Amer. Math. Soc. 138 (2010), no. 12, 4331– 4344.

    Google Scholar 

  7. Gao, Fuchang and Wellner, Jon A. Entropy estimate for high dimensional monotonic functions. J. Multivariate Anal. 98 (2007), no. 9, 1751–1764.

    Google Scholar 

  8. Gao, Fuchang and Wellner, Jon A. On the rate of convergence of the maximum likelihood estimator of a κ-monotone density. Sci. China Ser. A 52 (2009), no. 7, 1525–1538.

    Google Scholar 

  9. Gao, Fuchang and Wellner, Jon A. Global rates of convergence of the MLE for multivariate interval censoring. Technical Report, Department of Statistics, University of Washington (2012).

    Google Scholar 

  10. van de Geer, S. The entropy bound for monotone functions. Technical Report 9110, University of Leiden (1991).

    Google Scholar 

  11. Kuelbs, James and Li, Wenbo V. Metric entropy and the small ball problem for Gaussian measures. J. Funct. Anal. 116 (1993), no. 1, 133–157.

    Google Scholar 

  12. Nickl, Richard and Pötscher, Benedikt M. Bracketing metric entropy rates and empirical central limit theorems for function classes of Besov- and Sobolev-type. J. Theoret. Probab. 20 (2007), no. 2, 177–199.

    Google Scholar 

  13. van der Vaart, Aad W. Bracketing smooth functions. Stochastic Process. Appl. 52 (1994), no. 1, 93–105.

    Google Scholar 

  14. van der Vaart, Aad W. and Wellner, Jon A. Weak convergence and empirical processes. With applications to statistics. Springer Series in Statistics. Springer-Verlag, New York, 1996.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Idaho, 441103, Moscow, Idaho, 83844-1103, USA

    Fuchang Gao

Authors
  1. Fuchang Gao
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Correspondence to Fuchang Gao .

Editor information

Editors and Affiliations

  1. Georgia Institute of Technology, Atlanta, 30332, USA

    Christian Houdré

  2. University of Delaware, Newark, 19716, USA

    David M. Mason

  3. University of Tennessee, Knoxville, 37934, USA

    Jan Rosiński

  4. University of Washington, Seattle, 98195, USA

    Jon A. Wellner

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Gao, F. (2013). Bracketing Entropy of High Dimensional Distributions. In: Houdré, C., Mason, D., Rosiński, J., Wellner, J. (eds) High Dimensional Probability VI. Progress in Probability, vol 66. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0490-5_1

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