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Uniform-in-Bandwidth Functional Limit Laws

Author

Listed:
  • Paul Deheuvels

    (Université Pierre et Marie Curie (Paris 6))

  • Sarah Ouadah

    (Université Pierre et Marie Curie (Paris 6))

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.

Suggested Citation

  • Paul Deheuvels & Sarah Ouadah, 2013. "Uniform-in-Bandwidth Functional Limit Laws," Journal of Theoretical Probability, Springer, vol. 26(3), pages 697-721, September.
    • Handle: RePEc:spr:jotpro:v:26:y:2013:i:3:d:10.1007_s10959-011-0376-1
    DOI: 10.1007/s10959-011-0376-1
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    References listed on IDEAS

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    1. David Mason & Jan Swanepoel, 2011. "A general result on the uniform in bandwidth consistency of kernel-type function estimators," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(1), pages 72-94, May.
    2. Paul Deheuvels & David Mason, 2004. "General Asymptotic Confidence Bands Based on Kernel-type Function Estimators," Statistical Inference for Stochastic Processes, Springer, vol. 7(3), pages 225-277, October.
    3. Mason, David M., 1984. "A strong limit theorem for the oscillation modulus of the uniform empirical quantile process," Stochastic Processes and their Applications, Elsevier, vol. 17(1), pages 127-136, May.
    4. Einmahl, J.H.J. & Deheuvels, P., 2000. "Functional limit laws for the increments of Kaplan-Meier product-limit processes and applications," Other publications TiSEM ac9bbdc0-62f8-4b48-9a84-1, Tilburg University, School of Economics and Management.
    5. Deheuvels, Paul, 1992. "Functional laws of the iterated logarithm for large increments of empirical and quantile processes," Stochastic Processes and their Applications, Elsevier, vol. 43(1), pages 133-163, November.
    6. Uwe Einmahl & David M. Mason, 2000. "An Empirical Process Approach to the Uniform Consistency of Kernel-Type Function Estimators," Journal of Theoretical Probability, Springer, vol. 13(1), pages 1-37, January.
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