48 research outputs found
Kernel density estimation for stationary random fields
In this paper, under natural and easily verifiable conditions, we prove the
-convergence and the asymptotic normality of the
Parzen-Rosenblatt density estimator for stationary random fields of the form
, , where
are i.i.d real random variables and is a
measurable function defined on . Such kind of processes provides a
general framework for stationary ergodic random fields. A Berry-Esseen's type
central limit theorem is also given for the considered estimator.Comment: 25 page
Berry-Esseen's central limit theorem for non-causal linear processes in Hilbert space
Let be a real separable Hilbert space and a
sequence of bounded linear operators from to . We consider the linear
process defined for any in by
where
is a sequence of i.i.d. centered -valued
random variables. We investigate the rate of convergence in the CLT for and
in particular we obtain the usual Berry-Esseen's bound provided that
and
belongs to
On the central and local limit theorem for martingale difference sequences
Let (\Omega, \A, \mu) be a Lebesgue space and an ergodic measure
preserving automorphism on with positive entropy. We show that there
is a bounded and strictly stationary martingale difference sequence defined on
with a common non-degenerate lattice distribution satisfying the
central limit theorem with an arbitrarily slow rate of convergence and not
satisfying the local limit theorem. A similar result is established for
martingale difference sequences with densities provided the entropy is
infinite. In addition, the martingale difference sequence may be chosen to be
strongly mixing.Comment: Accepte pour publication dans Stochastics and Dynamic
Invariance principles for standard-normalized and self-normalized random fields
We investigate the invariance principle for set-indexed partial sums of a
stationary field of martingale-difference or
independent random variables under standard-normalization or self-normalization
respectively.Comment: Submitted for publicatio
Asymptotic normality of kernel estimates in a regression model for random fields
We establish the asymptotic normality of the regression estimator in a
fixed-design setting when the errors are given by a field of dependent random
variables. The result applies to martingale-difference or strongly mixing
random fields. On this basis, a statistical test that can be applied to image
analysis is also presented.Comment: 20 page
Kernel deconvolution estimation for random fields
In this work, we establish the asymptotic normality of the deconvolution
kernel density estimator in the context of strongly mixing random fields. Only
minimal conditions on the bandwidth parameter are required and a simple
criterion on the strong mixing coefficients is provided. Our approach is based
on the Lindeberg's method rather than on Bernstein's technique and coupling
arguments widely used in previous works on nonparametric estimation for spatial
processes. We deal also with nonmixing random fields which can be written as a
(nonlinear) functional of i.i.d. random fields by considering the physical
dependence measure coefficients introduced by Wu (2005).Comment: 28 pages. arXiv admin note: text overlap with arXiv:1109.269