卷积模型中线性泛函的自适应估计及其应用
Adaptive estimation of linear functionals in the convolution model and applications
摘要：
我们考虑模型$Z_i=X_i+\varepsilon_i$，对于i.i.d. $X_i$和$\varepsilon_i$以及独立序列$(X_i)_{i\in{\mathbb{N}}}$和$(\varepsilon_i)_{i\in{\mathbb{N}}}$。假设$\varepsilon_1$的密度$f_{\varepsilon}$是已知的，而$X_1$的密度（用$g$表示）是未知的。我们的目的是估计线性泛函的$g$，$<ψ,g>$为一个已知的功能$ψ$。我们提出了一个一般的估计$<ψ,g>$和研究其二次风险的收敛速度作为$g$，$f_{\varepsilon}$和$ψ$的光滑度的函数。还考虑了具有相关数据的不同背景，例如随机波动率和自回归相关异方差模型。然后，根据Laurent等人（Preprint（2006））在高斯白噪声模型中研究的方法，提出了一种自适应于未知$g$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。它们也适用于随机波动率模型的背景下。
英文摘要：
We consider the model $Z_i=X_i+\varepsilon_i$, for i.i.d. $X_i$'s and $\varepsilon_i$'s and independent sequences $(X_i)_{i\in{\mathbb{N}}}$ and $(\varepsilon_i)_{i\in{\mathbb{N}}}$. The density $f_{\varepsilon}$ of $\varepsilon_1$ is assumed to be known, whereas the one of $X_1$, denoted by $g$, is unknown. Our aim is to estimate linear functionals of $g$, $<ψ,g>$ for a known function $ψ$. We propose a general estimator of $<ψ,g>$ and study the rate of convergence of its quadratic risk as a function of the smoothness of $g$, $f_{\varepsilon}$ and $ψ$. 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 in the context of the stochastic volatility model.
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