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Uniform-in-Bandwidth Functional Limit Laws

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Abstract

We provide uniform-in-bandwidth functional limit laws for the increments of the empirical and quantile processes. Our theorems, established in the framework of convergence in probability, imply new sharp uniform-in-bandwidth limit laws for functional estimators. In particular, they yield the explicit value of the asymptotic limiting constant for the uniform-in-bandwidth sup-norm of the random error of kernel density estimators. We allow the bandwidth to vary within the complete range for which the estimators are consistent.

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Authors and Affiliations

  1. L.S.T.A., Université Pierre et Marie Curie (Paris 6), 7 avenue du Château, 92340, Bourg-la-Reine, France

    Paul Deheuvels

  2. L.S.T.A., Université Pierre et Marie Curie (Paris 6), T15-25, E2, 4 Place Jussieu, 75252, Paris Cedex 05, France

    Sarah Ouadah

Authors
  1. Paul Deheuvels
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  2. Sarah Ouadah
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Correspondence to Paul Deheuvels.

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Deheuvels, P., Ouadah, S. Uniform-in-Bandwidth Functional Limit Laws. J Theor Probab 26, 697–721 (2013). https://doi.org/10.1007/s10959-011-0376-1

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  • DOI: https://doi.org/10.1007/s10959-011-0376-1

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