991 research outputs found
Adaptive density estimation under dependence
Assume that is a real valued time series admitting a common
marginal density with respect to Lebesgue's measure. Donoho {\it et al.}
(1996) propose a near-minimax method based on thresholding wavelets to estimate
on a compact set in an independent and identically distributed setting. The
aim of the present work is to extend these results to general weak dependent
contexts. Weak dependence assumptions are expressed as decreasing bounds of
covariance terms and are detailed for different examples. The threshold levels
in estimators depend on weak dependence properties of the
sequence through the constant. If these properties are
unknown, we propose cross-validation procedures to get new estimators. These
procedures are illustrated via simulations of dynamical systems and non causal
infinite moving averages. We also discuss the efficiency of our estimators with
respect to the decrease of covariances bounds
Uniform limit theorems for the integrated periodogram of weakly dependent time series and their applications to Whittle's estimate
We prove uniform convergence results for the integrated periodogram of a
weakly dependent time series, namely a law of large numbers and a central limit
theorem. These results are applied to Whittle's parametric estimation. Under
general weak-dependence assumptions we derive uniform limit theorems and
asymptotic normality of Whittle's estimate for a large class of models. For
instance the causal -weak dependence property allows a new and unified
proof of those results for ARCH() and bilinear processes. Non causal
-weak dependence yields the same limit theorems for two-sided linear
(with dependent inputs) or Volterra processes
Sparsity considerations for dependent observations
The aim of this paper is to provide a comprehensive introduction for the
study of L1-penalized estimators in the context of dependent observations. We
define a general -penalized estimator for solving problems of
stochastic optimization. This estimator turns out to be the LASSO in the
regression estimation setting. Powerful theoretical guarantees on the
statistical performances of the LASSO were provided in recent papers, however,
they usually only deal with the iid case. Here, we study our estimator under
various dependence assumptions
Non-parametric estimation of time varying AR(1)--processes with local stationarity and periodicity
Extending the ideas of [7], this paper aims at providing a kernel based
non-parametric estimation of a new class of time varying AR(1) processes (Xt),
with local stationarity and periodic features (with a known period T), inducing
the definition Xt = at(t/nT)X t--1 + t for t N and with a t+T
at. Central limit theorems are established for kernel estima-tors
as(u) reaching classical minimax rates and only requiring low order moment
conditions of the white noise (t)t up to the second order
An invariance principle for weakly dependent stationary general models
The aim of this article is to refine a weak invariance principle for
stationary sequences given by Doukhan & Louhichi (1999). Since our conditions
are not causal our assumptions need to be stronger than the mixing and causal
-weak dependence assumptions used in Dedecker & Doukhan (2003). Here,
if moments of order exist, a weak invariance principle and convergence
rates in the CLT are obtained; Doukhan & Louhichi (1999) assumed the existence
of moments with order . Besides the previously used - and
-weak dependence conditions, we introduce a weaker one, ,
which fits the Bernoulli shifts with dependent inputs.Comment: 30 page
The notion of -weak dependence and its applications to bootstrapping time series
We give an introduction to a notion of weak dependence which is more general
than mixing and allows to treat for example processes driven by discrete
innovations as they appear with time series bootstrap. As a typical example, we
analyze autoregressive processes and their bootstrap analogues in detail and
show how weak dependence can be easily derived from a contraction property of
the process. Furthermore, we provide an overview of classes of processes
possessing the property of weak dependence and describe important probabilistic
results under such an assumption.Comment: Published in at http://dx.doi.org/10.1214/06-PS086 the Probability
Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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