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Exponential Bounds in Vapnik-Červonenkis Classes of Index 1

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Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference

Part of the book series: Progress in Probability ((PRPR,volume 30))

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Summary

For a probability measure P let P n be its empirical measures. The main result is that if there are constants C and γ such that for all P, all suitably measurable Vapnik-Cervonenkis classes C of index 1 containing the empty set and all M ≥ 1, we have

$$Pr(sup_{A\in C}n^{1/2}|(P_{n}-P)(A)|>M)\leq CM^{\gamma}exp(-2M^2), then\, \gamma\geq1$$

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Author information

Authors and Affiliations

  1. Department of Statistics, University of Georgia, Athens, GA, 30602, USA

    Daphne L. Smith

  2. Department of Mathematics, M.I.T., Cambridge, MA, 02139, USA

    R. M. Dudley

Authors
  1. Daphne L. Smith
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  2. R. M. Dudley
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Editor information

Editors and Affiliations

  1. Dept. of Mathematics, MIT, Cambridge, MA, 02139, USA

    Richard M. Dudley

  2. Department of Mathematics, Tufts University, Medford, MA, 02178, USA

    Marjorie G. Hahn

  3. Dept. of Mathematics, University of Wisconsin, Madison, WI, 53706, USA

    James Kuelbs

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Smith, D.L., Dudley, R.M. (1992). Exponential Bounds in Vapnik-Červonenkis Classes of Index 1. In: Dudley, R.M., Hahn, M.G., Kuelbs, J. (eds) Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference. Progress in Probability, vol 30. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0367-4_31

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  • DOI: https://doi.org/10.1007/978-1-4612-0367-4_31

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6728-7

  • Online ISBN: 978-1-4612-0367-4

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