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Online estimation of integrated squared density derivatives

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  • Mokkadem, Abdelkader
  • Pelletier, Mariane

Abstract

Hall and Marron (1987) introduced kernel estimators of integrals of the squared m-order derivatives of a probability density. Mokkadem and Pelletier (2020) gave recursive versions of their estimators, but the main drawback of these estimators is that their update requires the use of all past data. The aim of this paper is the study of online versions of the estimators introduced by Hall and Marron (1987), that is of estimators which are not only recursive, but which also have the property that their update uses only the last available data. Rates of convergence in mean squared error (MSE) are calculated. Similarly to the estimators of Hall and Marron (1987) and of Mokkadem and Pelletier (2020), our online estimators achieve the parametric rate n−1 when m=0 or when higher order kernels are used. For the case when the parametric rate is not obtained, we also study an online version of the estimator proposed by Jones and Sheather (1991). Finally, we provide recursive estimators of the optimal bandwidth in the framework of density estimation.

Suggested Citation

  • Mokkadem, Abdelkader & Pelletier, Mariane, 2020. "Online estimation of integrated squared density derivatives," Statistics & Probability Letters, Elsevier, vol. 166(C).
    • Handle: RePEc:eee:stapro:v:166:y:2020:i:c:s0167715220301838
    DOI: 10.1016/j.spl.2020.108880
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    References listed on IDEAS

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    1. Hall, Peter & Marron, J. S., 1987. "Estimation of integrated squared density derivatives," Statistics & Probability Letters, Elsevier, vol. 6(2), pages 109-115, November.
    2. Evarist Giné & David M. Mason, 2008. "Uniform in Bandwidth Estimation of Integral Functionals of the Density Function," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(4), pages 739-761, December.
    3. Jones, M. C. & Sheather, S. J., 1991. "Using non-stochastic terms to advantage in kernel-based estimation of integrated squared density derivatives," Statistics & Probability Letters, Elsevier, vol. 11(6), pages 511-514, June.
    4. Godichon-Baggioni, Antoine, 2016. "Estimating the geometric median in Hilbert spaces with stochastic gradient algorithms: Lp and almost sure rates of convergence," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 209-222.
    5. Anton Schick & Wolfgang Wefelmeyer, 2008. "Root-n consistency in weighted L 1 -spaces for density estimators of invertible linear processes," Statistical Inference for Stochastic Processes, Springer, vol. 11(3), pages 281-310, October.
    6. Mokkadem, Abdelkader & Pelletier, Mariane, 2020. "Recursive estimators of integrated squared density derivatives," Statistics & Probability Letters, Elsevier, vol. 157(C).
    7. Anton Schick & Wolfgang Wefelmeyer, 2004. "Root n consistent and optimal density estimators for moving average processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 31(1), pages 63-78, March.
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