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References (15)
-
Richard Nickl, B. Pötscher (2007)
Bracketing Metric Entropy Rates and Empirical Central Limit Theorems for Function Classes of Besov- and Sobolev-TypeJournal of Theoretical Probability, 20
-
英敦 塚原 (2009)
Aad W. van der Vaart and Jon A. Wellner: Weak Convergence and Empirical Processes: With Applications to Statistics, Springer,1996年,xvi + 508ページ., 61
-
J. Doob (1953)
Stochastic processes -
Fuchang Gao, J. Wellner (2012)
Global rates of convergence of the MLE for multivariate interval censoring.Electronic journal of statistics, 7
-
Fuchang Gao (2008)
Entropy Estimate for $k$-Monotone Functions via Small Ball Probability of Integrated Brownian MotionsElectronic Communications in Probability, 13
-
D. Bilyk, M. Lacey, A. Vagharshakyan (2007)
On the Small Ball Inequality in All DimensionsarXiv: Classical Analysis and ODEs
-
Fuchang Gao, Wenbo Li, J. Wellner (2010)
How many Laplace transforms of probability measures are there?Proceedings of the American Mathematical Society. American Mathematical Society, 138 12
-
T. Dunker, W. Linde, T. Kühn, M. Lifshits (1999)
Metric Entropy of Integration Operators and Small Ball Probabilities for the Brownian SheetJournal of Approximation Theory, 101
-
Fuchang Gao, J. Wellner (2005)
Entropy estimate for high-dimensional monotonic functionsJournal of Multivariate Analysis, 98
-
William Chen (1980)
On irregularities of distribution.Mathematika, 27
-
A. Vaart (1994)
Bracketing smooth functionsStochastic Processes and their Applications, 52
-
Fuchang Gao, J. Wellner (2009)
On the rate of convergence of the maximum likelihood estimator of a k-monotone densityScience in China Series A: Mathematics, 52
-
T. Mikosch, A. Vaart, J. Wellner (1996)
Weak Convergence and Empirical Processes: With Applications to Statistics -
J. Kuelbs, Wenbo Li (1993)
Metric entropy and the small ball problem for Gaussian measuresJournal of Functional Analysis, 116
-
R. Blei, Fuchang Gao, Wenbo Li (2007)
Metric entropy of high dimensional distributions, 135
- Publisher
- Springer Basel
- Copyright
- © Springer Basel 2013
- ISBN
- 978-3-0348-0489-9
- Pages
- 3 –17
- DOI
- 10.1007/978-3-0348-0490-5_1
- Publisher site
- See Chapter on Publisher Site
Abstract
[Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{F}_{d}$$ \end{document} be the class of probability distribution functions on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$[0,\,1]^{d},\,{d}\geq{2}$$ \end{document}. The following estimate for the bracketing entropy of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{F}_{d}$$ \end{document} in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$[L]^{p}$$\end{document} norm, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$1\,\leq\,p\,{<} \infty $$ \end{document}, is obtained: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\rm log}{N_{[\,]}}(\varepsilon, \mathcal{F}_{d},{\parallel.\parallel}_p)=O(\varepsilon^{-1}{|\rm log\varepsilon|^{2(\rm d-1)}}).$$ \end{document} Based on this estimate, a general relation between bracketing entropy in the Lp norm and metric entropy in the L1 norm for multivariate smooth functions is established.]
Published: Apr 1, 2013
Keywords: Bracketing entropy; metric entropy; high dimensional distribution