Uniform Entropy Numbers

  • Aad W. van der Vaart
  • Jon A. Wellner
Part of the Springer Series in Statistics book series (SSS)


In Section 2.5.1 the empirical process was shown to converge weakly for indexing sets F satisfying a uniform entropy condition. In particular, if
$$ s\mathop u\limits_Q p\log N\left( {\varepsilon \parallel F{\parallel _{Q,2}},F,\mathop L\nolimits_2 \left( Q \right)} \right) \leqslant K{\left( {\frac{1}{\varepsilon }} \right)^{2 - \delta }} $$
for some δ >0, then the entropy integral (2.5.1) converges and F is a Donsker class for any probability measure P such that P*F 2 < ∞, provided measurability conditions are met. Many classes of functions satisfy this condition and often even the much stronger condition
$$ s\mathop u\limits_Q pN\left( {\varepsilon \parallel F{\parallel _{Q,2}},F,{L_2}\left( Q \right)} \right) \leqslant K{\left( {\frac{1}{\varepsilon }} \right)^v},0 < \varepsilon < 1 $$
for some number V. In this chapter this is shown for classes satisfying certain combinatorial conditions. For classes of sets, these were first studied by Vapnik and Červonenkis, whence the name VC-classes. In the second part of this chapter, VC-classes of functions are defined in terms of VC-classes of sets. The remainder of this chapter considers operations on classes that preserve entropy properties, such as taking convex hulls.


Probability Measure Convex Hull Empirical Process Symmetric Convex Entropy Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • Aad W. van der Vaart
    • 1
  • Jon A. Wellner
    • 2
  1. 1.Department of Mathematics and Computer ScienceFree UniversityAmsterdamThe Netherlands
  2. 2.StatisticsUniversity of WashingtonSeattleUSA

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