1,193 research outputs found
Adaptive estimation of linear functionals in the convolution model and applications
We consider the model , for i.i.d. 's and
's and independent sequences and
. The density of
is assumed to be known, whereas the one of , denoted by
, is unknown. Our aim is to estimate linear functionals of ,
for a known function $\psi$. We propose a general estimator of and
study the rate of convergence of its quadratic risk as a function of the
smoothness of , and . Different contexts with
dependent data, such as stochastic volatility and AutoRegressive Conditionally
Heteroskedastic models, are also considered. An estimator which is adaptive to
the smoothness of unknown is then proposed, following a method studied by
Laurent et al. (Preprint (2006)) in the Gaussian white noise model. We give
upper bounds and asymptotic lower bounds of the quadratic risk of this
estimator. The results are applied to adaptive pointwise deconvolution, in
which context losses in the adaptive rates are shown to be optimal in the
minimax sense. They are also applied in the context of the stochastic
volatility model.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ146 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Non-archimedean Yomdin-Gromov parametrizations and points of bounded height
We prove an analogue of the Yomdin-Gromov Lemma for -adic definable sets
and more broadly in a non-archimedean, definable context. This analogue keeps
track of piecewise approximation by Taylor polynomials, a nontrivial aspect in
the totally disconnected case. We apply this result to bound the number of
rational points of bounded height on the transcendental part of -adic
subanalytic sets, and to bound the dimension of the set of complex polynomials
of bounded degree lying on an algebraic variety defined over , in analogy to results by Pila and Wilkie, resp. by Bombieri and Pila.
Along the way we prove, for definable functions in a general context of
non-archimedean geometry, that local Lipschitz continuity implies piecewise
global Lipschitz continuity.Comment: 54 pages; revised, section 5.6 adde
Variational approach to the excitonic phase transition in graphene
We analyze the Coulomb interacting problem in undoped graphene layers by
using an excitonic variational ansatz. By minimizing the energy, we derive a
gap equation which reproduces and extends known results. We show that a full
treatment of the exchange term, which includes the renormalization of the Fermi
velocity, tends to suppress the phase transition by increasing the critical
coupling at which the excitonic instability takes place.Comment: 4 page
Anisotropic adaptive kernel deconvolution
International audienceIn this paper, we consider a multidimensional convolution model for which we provide adaptive anisotropic kernel estimators of a signal density measured with additive error. For this, we generalize Fan's~(1991) estimators to multidimensional setting and use a bandwidth selection device in the spirit of Goldenschluger and Lepski's~(2011) proposal fr density estimation without noise. We consider first the pointwise setting and then, we study the integrated risk. Our estimators depend on an automatically selected random bandwidth. We assume both ordinary and super smooth components for measurement errors, which have known density. We also consider both anisotropic H\"{o}lder and Sobolev classes for . We provide non asymptotic risk bounds and asymptotic rates for the resulting data driven estimator, which is proved to be adaptive. We provide an illustrative simulation study, involving the use of Fast Fourier Transform algorithms. We conclude by a proposal of extension of the method to the case of unknown noise density, when a preliminary pure noise sample is available
Nonparametric adaptive estimation for pure jump Lévy processes
International audienceThis paper is concerned with nonparametric estimation of the Lévy density of a pure jump Lévy process. The sample path is observed at discrete instants with fixed sampling interval. We construct a collection of estimators obtained by deconvolution methods and deduced from appropriate estimators of the characteristic function and its first derivative. We obtain a bound for the -risk, under general assumptions on the model. Then we propose a penalty function that allows to build an adaptive estimator. The risk bound for the adaptive estimator is obtained under additional assumptions on the Lévy density. Examples of models fitting in our framework are described and rates of convergence of the estimator are discussed
Asymptotic Theory for Multivariate GARCH Processes
We provide in this paper asymptotic theory for the multivariate GARCH (p,q) process. Strong consistency of the quasi-maximum likelihood estimator (MLE) is established by appealing to conditions given in Jeantheau [19] in conjunction with a result given by Boussama [9] concerning the existence of a stationary and ergodic solution to the multivariate GARCH (p,q) process. We prove asymptotic normality of the quasi-MLE when the initial state is either stationary or fixed.Asymptotic normality, BEKK, consistency, GARCH, Martingale CLT
Asymptotic theory for multivariate GARCH processes
AbstractWe provide in this paper asymptotic theory for the multivariate GARCH(p,q) process. Strong consistency of the quasi-maximum likelihood estimator (MLE) is established by appealing to conditions given by Jeantheau (Econometric Theory 14 (1998), 70) in conjunction with a result given by Boussama (Ergodicity, mixing and estimation in GARCH models, Ph.D. Dissertation, University of Paris 7, 1998) concerning the existence of a stationary and ergodic solution to the multivariate GARCH(p,q) process. We prove asymptotic normality of the quasi-MLE when the initial state is either stationary or fixed
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