• 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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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.

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

First available in Project Euclid: 23 August 2004

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integrated squared deviation kernel density estimator law of the iterated logarithm


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.

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  • [1] Beirlant, J. and Mason, D.M. (1995) On the asymptotic normality of Lp-norms of empirical functionals. Math. Methods Statist., 4, 1-19.
  • [2] Bickel, P.J. and Rosenblatt, M. (1973) On some global measures of the deviations of density estimation. Ann. Statist., 1, 1071-1095.
  • [3] Csörgö, M. and Horváth, L. (1988) Central limit theorems for Lp-norms of density estimators. Z. Wahrscheinlichkeitstheorie Verw. Geb., 80, 269-291.
  • [4] de la Penã, V. and Giné, E. (1999) Decoupling: From Dependence to Independence. New York:, Springer-Verlag.
  • [5] de la Penã, V. and Montgomery-Smith, S. (1994) Bounds for the tail probabilities of U-statistics and quadratic forms. Bull. Amer. Math. Soc., 31, 223-227.
  • [6] Dunford, N. and Schwartz, J.T. (1964) Linear Operators, Part II, 2nd printing. New York:, Wiley.
  • [7] Eggermont, P.P.B. and LaRiccia, V.N. (2001) Maximum Penalized Likelihood Estimation, Volume 1: Density Estimation. New York:, Springer-Verlag.
  • [8] Fernique, X. (1970) Integrabilité des vecteurs gaussiens. C. R. Acad. Sci. Paris, Sér. A, 270, 1698-, 1699.
  • [9] Giné, E., Latala, R. and Zinn, J. (2000) Exponential and moment inequalities for U-statistics. In E. Giné, D.M. Mason and J.A. Wellner (eds), High Dimensional Probability II, Progr. Probab. 47, pp. 13-38. Boston:, Birkhäuser.
  • [10] Giné, E., Mason, D.M. and Zaitsev, A. Yu. (2003) The L1-norm density estimation process. Ann. Probab., 31, 719-768.
  • [11] Hall, P. (1984) Central limit theorem for integrated square error of multivariate nonparametric density estimators. J. Multivariate Anal., 14, 1-16.
  • [12] Komlo´s, J., Major, P. and Tusnády, G. (1975) An approximation of partial sums of independent rv´s and the sample distribution function, I. Z. Wahrscheinlichkeitstheorie Verw. Geb., 32, 111-131.
  • [13] Ledoux, M. and Talagrand, M. (1991) Probability in Banach Spaces. Berlin:, Springer-Verlag.
  • [14] Mason, D.M. (2003) Representations for estimators of integral functionals of the density function. Austrian J. Statist., 32, 131-142., Abstract can also be found in the ISI/STMA publication
  • [15] Montgomery-Smith, S.J. (1993) Comparison of sums of independent identically distributed random vectors. Probab. Math. Statist., 14, 281-285.
  • [16] Nadaraya, N.A. (1989) Nonparametric Estimation of Probability Densities and Regression Curves. Amsterdam:, Kluwer.
  • [17] Pinsky, M. (1966) An elementary derivation of Khintchinés estimate for large deviations. Proc. Amer. Math. Soc., 22, 288-290.
  • [18] Rosenblatt, M. (1975) A quadratic measure of the deviation of two-dimensional density estimates and a test of independence. Ann. Statist., 3, 1-14.
  • [19] Shorack, G. and Wellner, J. (1986) Empirical Processes with Applications to Statistics. New York:, Wiley.