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...
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 ....
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
....
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.
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