The law of the iterated logarithm for the multivariate kernel mode estimator

Abdelkader Mokkadem; Mariane Pelletier

ESAIM: Probability and Statistics (2010)

  • Volume: 7, page 1-21
  • ISSN: 1292-8100

Abstract

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Let θ be the mode of a probability density and θn its kernel estimator. In the case θ is nondegenerate, we first specify the weak convergence rate of the multivariate kernel mode estimator by stating the central limit theorem for θn - θ. Then, we obtain a multivariate law of the iterated logarithm for the kernel mode estimator by proving that, with probability one, the limit set of the sequence θn - θ suitably normalized is an ellipsoid. We also give a law of the iterated logarithm for the lp norms, p ∈ [1,∞], of θn - θ. Finally, we consider the case θ is degenerate and give the exact weak and strong convergence rate of θn - θ in the univariate framework.

How to cite

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Mokkadem, Abdelkader, and Pelletier, Mariane. "The law of the iterated logarithm for the multivariate kernel mode estimator." ESAIM: Probability and Statistics 7 (2010): 1-21. <http://eudml.org/doc/104303>.

@article{Mokkadem2010,
abstract = { Let θ be the mode of a probability density and θn its kernel estimator. In the case θ is nondegenerate, we first specify the weak convergence rate of the multivariate kernel mode estimator by stating the central limit theorem for θn - θ. Then, we obtain a multivariate law of the iterated logarithm for the kernel mode estimator by proving that, with probability one, the limit set of the sequence θn - θ suitably normalized is an ellipsoid. We also give a law of the iterated logarithm for the lp norms, p ∈ [1,∞], of θn - θ. Finally, we consider the case θ is degenerate and give the exact weak and strong convergence rate of θn - θ in the univariate framework. },
author = {Mokkadem, Abdelkader, Pelletier, Mariane},
journal = {ESAIM: Probability and Statistics},
keywords = {Density; mode; kernel estimator; central limit theorem; law of the iterated logarithm.; density; law of the iterated logarithm},
language = {eng},
month = {3},
pages = {1-21},
publisher = {EDP Sciences},
title = {The law of the iterated logarithm for the multivariate kernel mode estimator},
url = {http://eudml.org/doc/104303},
volume = {7},
year = {2010},
}

TY - JOUR
AU - Mokkadem, Abdelkader
AU - Pelletier, Mariane
TI - The law of the iterated logarithm for the multivariate kernel mode estimator
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 7
SP - 1
EP - 21
AB - Let θ be the mode of a probability density and θn its kernel estimator. In the case θ is nondegenerate, we first specify the weak convergence rate of the multivariate kernel mode estimator by stating the central limit theorem for θn - θ. Then, we obtain a multivariate law of the iterated logarithm for the kernel mode estimator by proving that, with probability one, the limit set of the sequence θn - θ suitably normalized is an ellipsoid. We also give a law of the iterated logarithm for the lp norms, p ∈ [1,∞], of θn - θ. Finally, we consider the case θ is degenerate and give the exact weak and strong convergence rate of θn - θ in the univariate framework.
LA - eng
KW - Density; mode; kernel estimator; central limit theorem; law of the iterated logarithm.; density; law of the iterated logarithm
UR - http://eudml.org/doc/104303
ER -

References

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