Adaptive estimation of linear functionals in the convolution model and applications
Adaptive estimation of linear functionals in the convolution model and applications
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DOI：
10.3150/08-bej146
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发表时间：
2009-02-01
期刊：
影响因子：
1.5
通讯作者：
Comte, F.
中科院分区：
文献类型：
--
作者：
Butucea, C.;Comte, F.
本文考虑模型Z（i）= X-i + X（i），其中i. X-i's和n（i）'s以及独立序列（X-i）（i <$N）和（n（i））（i <$N）。假设n（1）的密度f（n）是已知的，而X-1的密度g是未知的。我们的目标是估计线性泛函g，为一个已知的功能psi。本文提出了（psi，g）的一个一般估计，并研究了它的二次风险的收敛速度作为g，f（ψ）的光滑性的函数，同时考虑了随机波动率和自回归相关异方差模型等不同的相依数据情形.在高斯白色噪声模型下，根据Laurent等人（Preprint（2006））研究的方法，提出了一种自适应于未知g的平滑性的估计器。给出了该估计的二次风险的上界和渐近下界。结果被施加到自适应逐点反卷积，其中上下文损失的自适应率被证明是最佳的最小最大意义。它们也被应用于随机波动率模型的背景下。
We consider the model Z(i) = X-i + epsilon(i), for i.i.d. X-i's and epsilon(i)'s and independent sequences (X-i)(i epsilon N) and (epsilon(i))(i epsilon N). The density f(epsilon) of epsilon(1) is assumed to be known, whereas the one of X-1, denoted by g, is unknown. Our aim is to estimate linear functionals of g, for a known function psi. We propose a general estimator of (psi, g) and study the rate of convergence of its quadratic risk as a function of the smoothness of g, f(epsilon) 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 g 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 ill the context of the stochastic volatility model.