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On Continuity and Strict Increase of the CDF for the Sup-Functional of a Gaussian Process with Applications to Statistics

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Abstract.

We describe in a coherent way a proof that the cdf of the supremum of a mean-zero Gaussian process is continuous and strictly increasing as soon as it is not degenerate at zero. The importance of both properties is illustrated by an application to statistics in connection with constructing confidence bands for unknown probability distributions indexed by Vapnik–Chervonenkis classes of sets in arbitrary sample spaces via bootstrapping.

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Authors and Affiliations

  1. Mathematical Institute, University of Munich, Theresienstr. 39, D-80333, Munich, Germany

    Peter Gaenssler & Daniel Rost

  2. Eichenstr. 29 A, D-82347, Bernried, Germany

    Peter Gaenssler

  3. Centre for Mathematical Biology, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    Péter Molnár

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  1. Peter Gaenssler
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  2. Péter Molnár
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  3. Daniel Rost
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Received: October 10, 2006. Revised: April 19, 2007.

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Gaenssler, P., Molnár, P. & Rost, D. On Continuity and Strict Increase of the CDF for the Sup-Functional of a Gaussian Process with Applications to Statistics. Result. Math. 51, 51–60 (2007). https://doi.org/10.1007/s00025-007-0257-1

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  • DOI: https://doi.org/10.1007/s00025-007-0257-1

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