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Proving consistency of non-standard kernel estimators

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Abstract

We develop general methods based upon empirical process techniques to prove uniform in bandwidth consistency of a class of non-standard kernel-type function estimators. Examples include projection pursuit regression and conditional distribution estimation. Our results are especially useful to establish uniform consistency of data-driven bandwidth kernel-type function estimators.

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References

  • Deheuvels P (1991) Laws of the iterated logarithm for density estimations. In: Roussas G (ed) Nonparametric functional estimation and related topics. NATO ASI Series C, Mathematics and Physical Sciences, vol 335. Kluwer, Dordrecht, pp 19–29

    Google Scholar 

  • Deheuvels P, Mason DM (1992) Functional laws of the iterated logarithm for the increments of empirical and quantile processes. Ann Probab 20: 1248–1287

    Article  MathSciNet  MATH  Google Scholar 

  • Deheuvels P, Mason DM (2004) General asymptotic confidence bands based on kernel-type function estimators. Stat Inference Stoch Process 7: 225–277

    Article  MathSciNet  MATH  Google Scholar 

  • Dony J (2008) Nonparametric regression estimation: an empirical process approach to the uniform in bandwidth consistency of kernel-type estimators and conditional U-statistics. PhD Thesis, Vrije Universiteit Brussel, Belgium

  • Dony J, Mason DM (2008) Uniform in bandwidth consistency of conditional U-statistics. Bernoulli 4: 1108–1133

    Article  MathSciNet  Google Scholar 

  • Dony J, Mason DM (2010) Uniform in bandwidth consistency of kernel estimators of the tail index. Extremes 13: 353–371

    Article  MathSciNet  MATH  Google Scholar 

  • Dony J, Einmahl U, Mason DM (2006) Uniform in bandwidth consistency of local polynomial regression function estimators. Aust J Stat 35: 105–120

    Google Scholar 

  • Einmahl U, Mason DM (2000) An empirical process approach to the uniform consistency of kernel-type function estimators. J Theor Probab 13: 1–37

    Article  MathSciNet  MATH  Google Scholar 

  • Einmahl U, Mason DM (2005) Uniform in bandwidth consistency of kernel-type function estimators. Ann Stat 33: 1380–1403

    Article  MathSciNet  MATH  Google Scholar 

  • Giné E, Sang H (2010) Uniform asymptotics for kernel density estimators with variable bandwidths. J Nonpar Stat 22: 773–795

    Article  MATH  Google Scholar 

  • Hall P (1989) On projection pursuit regression. Ann Stat 17: 573–588

    Article  MATH  Google Scholar 

  • Hall P, Yao Q (2005) Approximating conditional distribution functions using dimension reduction. Ann Stat 33: 1404–1421

    Article  MathSciNet  MATH  Google Scholar 

  • Hoffmann-Jørgensen J (1994) Probability with a view toward statistics, vol 2. Chapman & Hall/CRC Probability Series, Boca Raton

    Google Scholar 

  • Mason DM, Swanepoel J (2011) A general result on the uniform in bandwidth consistency of kernel-type function estimators. Test. 20: 72–94

    Article  MathSciNet  Google Scholar 

  • Nolan D, Marron JS (1989) Uniform consistency of automatic and location-adaptive delta-sequence estimators. Probab Theory Relat Fields 80: 619–632

    Article  MathSciNet  MATH  Google Scholar 

  • Talagrand M. (1994) Sharper bounds for Gaussian and empirical processes. Ann Probab 22: 28–76

    Article  MathSciNet  MATH  Google Scholar 

  • van der Vaart AW, Wellner JA (1996) Weak convergence and empirical processes. With applications to statistics. Springer Series in Statistics. Springer, New York

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Authors and Affiliations

  1. Department of Food and Resource, Economics, University of Delaware, Newark, DE, 19717, USA

    David M. Mason

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  1. David M. Mason
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Correspondence to David M. Mason.

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Mason, D.M. Proving consistency of non-standard kernel estimators. Stat Inference Stoch Process 15, 151–176 (2012). https://doi.org/10.1007/s11203-012-9068-4

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  • DOI: https://doi.org/10.1007/s11203-012-9068-4

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