Bernoulli

  • Bernoulli
  • Volume 10, Number 4 (2004), 721-752.

The law of the iterated logarithm for the integrated squared deviation of a kernel density estimator

Evarist Giné and David M. Mason

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Abstract

Let f n ,K denote a kernel estimator of a density f in R such that R f p(x)dx< for some p>2. It is shown, under quite general conditions on the kernel K and on the window sizes, that the centred integrated squared deviation of f n ,K from its mean, f n ,K-Ef n ,K 2 2-Ef n ,K-Ef n ,K 2 2 satisfies a law of the iterated logarithm (LIL). This is then used to obtain an LIL for the deviation from the true density, f n ,K-f 2 2-Ef n ,K-f 2 2 . The main tools are the Komlós-Major-Tusnády approximation, a moderate-deviation result for triangular arrays of weighted chi-square variables adapted from work by Pinsky, and an exponential inequality due to Giné, Latała and Zinn for degenerate U-statistics applied in combination with decoupling and maximal inequalities.

Article information

Source
Bernoulli, Volume 10, Number 4 (2004), 721-752.

Dates
First available in Project Euclid: 23 August 2004

Permanent link to this document
https://projecteuclid.org/euclid.bj/1093265638

Digital Object Identifier
doi:10.3150/bj/1093265638

Mathematical Reviews number (MathSciNet)
MR2076071

Zentralblatt MATH identifier
1067.62048

Keywords
integrated squared deviation kernel density estimator law of the iterated logarithm

Citation

Giné, Evarist; Mason, David M. The law of the iterated logarithm for the integrated squared deviation of a kernel density estimator. Bernoulli 10 (2004), no. 4, 721--752. doi:10.3150/bj/1093265638. https://projecteuclid.org/euclid.bj/1093265638


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