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Moderate Deviations and Large Deviations for Kernel Density Estimators

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  • Fuqing Gao

    (Wuhan University)

Abstract

Let f n be the non-parametric kernel density estimator based on a kernel function K and a sequence of independent and identically distributed random variables taking values in ℝ d . It is proved that if the kernel function is an integrable function with bounded variation, and the common density function f of the random variables is continuous and f(x) → 0 as |x| → ∞, then the moderate deviation principle and large deviation principle for $$\{ \sup _{x \in \mathbb{R}^d } |f_n (x) - E(f_n (x))|,n \geqslant 1\} $$ hold.

Suggested Citation

  • Fuqing Gao, 2003. "Moderate Deviations and Large Deviations for Kernel Density Estimators," Journal of Theoretical Probability, Springer, vol. 16(2), pages 401-418, April.
    • Handle: RePEc:spr:jotpro:v:16:y:2003:i:2:d:10.1023_a:1023574711733
    DOI: 10.1023/A:1023574711733
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    References listed on IDEAS

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    1. 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.
    2. 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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    Cited by:

    1. Bitseki Penda, S. Valère, 2023. "Moderate deviation principles for kernel estimator of invariant density in bifurcating Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 282-314.
    2. 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.
    3. Siyu Liu & Xiequan Fan & Haijuan Hu & Paul Doukhan, 2024. "Pointwise Sharp Moderate Deviations for a Kernel Density Estimator," Mathematics, MDPI, vol. 12(20), pages 1-9, October.

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