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Large Deviations Limit Theorems for the Kernel Density Estimator

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  • Djamal Louani

Abstract

We establish pointwise and uniform large deviations limit theorems of Chernoff‐type for the non‐parametric kernel density estimator based on a sequence of independent and identically distributed random variables. The limits are well‐identified and depend upon the underlying kernel and density function. We derive then some implications of our results in the study of asymptotic efficiency of the goodness‐of‐fit test based on the maximal deviation of the kernel density estimator as well as the inaccuracy rate of this estimate

Suggested Citation

  • Djamal Louani, 1998. "Large Deviations Limit Theorems for the Kernel Density Estimator," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 25(1), pages 243-253, March.
    • Handle: RePEc:bla:scjsta:v:25:y:1998:i:1:p:243-253
    DOI: 10.1111/1467-9469.00101
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    Cited by:

    1. Djamal Louani, 2005. "UniformL 1 -distance large deviations in nonparametric density estimation," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 14(1), pages 75-98, June.
    2. Diallo, Amadou Oury Korbe & Louani, Djamal, 2013. "Moderate and large deviation principles for the hazard rate function kernel estimator under censoring," Statistics & Probability Letters, Elsevier, vol. 83(3), pages 735-743.
    3. Gao, Fuqing, 2008. "Moderate deviations and law of the iterated logarithm in for kernel density estimators," Stochastic Processes and their Applications, Elsevier, vol. 118(3), pages 452-473, March.
    4. Fuqing Gao, 2003. "Moderate Deviations and Large Deviations for Kernel Density Estimators," Journal of Theoretical Probability, Springer, vol. 16(2), pages 401-418, April.
    5. Korostelev Alexander, 2003. "The Bahadur risk in probability density estimation," Statistics & Risk Modeling, De Gruyter, vol. 21(2/2003), pages 139-148, February.
    6. Djamal Louani & Sidi Mohamed Ould Maouloud, 2012. "Some Functional Large Deviations Principles in Nonparametric Function Estimation," Journal of Theoretical Probability, Springer, vol. 25(1), pages 280-309, March.
    7. Osmoukhina, Anna V., 2001. "Large deviations probabilities for a test of symmetry based on kernel density estimator," Statistics & Probability Letters, Elsevier, vol. 54(4), pages 363-371, October.
    8. Kakizawa, Yoshihide, 2007. "Moderate deviations for quadratic forms in Gaussian stationary processes," Journal of Multivariate Analysis, Elsevier, vol. 98(5), pages 992-1017, May.
    9. Bitseki Penda, S. Valère & Olivier, Adélaïde, 2018. "Moderate deviation principle in nonlinear bifurcating autoregressive models," Statistics & Probability Letters, Elsevier, vol. 138(C), pages 20-26.
    10. Cyrille Joutard, 2013. "Strong large deviations for arbitrary sequences of random variables," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(1), pages 49-67, February.
    11. Song, Weixing, 2010. "Moderate deviations for deconvolution kernel density estimators with ordinary smooth measurement errors," Statistics & Probability Letters, Elsevier, vol. 80(3-4), pages 169-176, February.
    12. Pablo Martínez-Camblor & Jacobo Uña-Álvarez, 2013. "Studying the bandwidth in $$k$$ -sample smooth tests," Computational Statistics, Springer, vol. 28(2), pages 875-892, April.
    13. Cyrille Joutard, 2002. "Sharp Large Deviations in Nonparametric Estimation," Working Papers 2002-43, Center for Research in Economics and Statistics.

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