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Large and Moderate Deviations Principles for Recursive Kernel Estimator of a Multivariate Density and its Partial Derivatives

Mokkadem, Abdelkader;  Mariane, Pelletier;  Baba, Thiam — 2006

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 62G07, 60F10. In this paper we prove large and moderate deviations principles for the recursive kernel estimator of a probability density function and its partial derivatives. Unlike the density estimator, the derivatives estimators exhibit a quadratic behaviour not only for the moderate deviations scale but also for the large deviations one. We provide results both for the pointwise and the uniform deviations.

On the Recursive Estimation of the Location and of the Size of the Mode of a Probability Density

Djeddour, Khédidja;  Mokkadem, Abdelkader;  Pelletier, Mariane — 2008

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 62G07, 62L20. Tsybakov [31] introduced the method of stochastic approximation to construct a recursive estimator of the location q of the mode of a probability density. The aim of this paper is to provide a companion algorithm to Tsybakov's algorithm, which allows to simultaneously recursively approximate the size m of the mode. We provide a precise study of the joint weak convergence rate of both estimators. Moreover, we introduce the averaging...

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

Abdelkader Mokkadem;  Mariane Pelletier — 2003

ESAIM: Probability and Statistics

Let θ be the mode of a probability density and θ n 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 θ n - 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 θ n - θ suitably normalized is an ellipsoid. We also give a law of the iterated logarithm for the l p norms, p ∈ [ 1 , ∞ ] , of θ n - θ ....

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

Abdelkader Mokkadem;  Mariane Pelletier — 2010

ESAIM: Probability and Statistics

Let be the mode of a probability density and its kernel estimator. In the case is nondegenerate, we first specify the weak convergence rate of the multivariate kernel mode estimator by stating the central limit theorem for . 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 suitably normalized is an ellipsoid. We also give a law of the iterated logarithm for the norms, , of ....

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