Minimax theory of estimation of linear functionals of the deconvolution density with or without sparsity
Minimax theory of estimation of linear functionals of the deconvolution density with or without sparsity
复制标题
具有或不具有稀疏性的反卷积密度线性函数估计的极小极大理论
DOI：
10.1214/16-aos1498
复制
发表时间：
2014
期刊：
影响因子：
--
通讯作者：
M. Pensky
中科院分区：
文献类型：
--
作者：
M. Pensky
本文在I.I.D.的基础上考虑了估计未知反卷积密度的线性泛函的问题。观测结果$Y_i=\theta_i+\xi_i$，其中$\xi_i$具有已知的pdf$g$，$f$是$\theta_i$的pdf。尽管许多作者对这一问题的各个方面和具体情况进行了论述，但仍有许多空白。特别地，对于任意函数$\varphi$，$\Phi$的估计量没有极小极大下界。一般的风险上界仅涵盖存在$\varphi$的傅里叶变换的情况。此外，当观测向量稀疏时，没有理论来估计$\Phi$。另外，到目前为止，间接观测中泛函的估计问题一直被当作一个独立的问题来处理，与泛函的估计无关。本文的目的是填补这一空白，发展一般的极大极小估计理论。当函数$varphi$是平方可积时，我们给出了估计$Phi$(和$Phi_n$)的一般方法，并给出了风险的上界和极小下界。此外，我们将该理论推广到不存在$\varphi$的傅里叶变换，并且$\phi$可以表示为$f$及其导数的傅里叶变换的线性泛函的情况。最后，我们推广了我们的结果来处理向量$\theta$稀疏的情况。作为该理论的直接应用，我们得到了许多新的结果，并自动恢复了已有的各种问题的结果，例如估计反褶积密度的$(2m+1)$绝对矩或广义矩，估计混合CDF或估计具有经典和Berkson误差的混合pdf。
The present paper considers a problem of estimating a linear functional $\Phi=\int_{-\infty}^\infty \varphi(x) f(x)dx$ of an unknown deconvolution density $f$ on the basis of i.i.d. observations $Y_i = \theta_i + \xi_i$ where $\xi_i$ has a known pdf $g$ and $f$ is the pdf of $\theta_i$. Although various aspects and particular cases of this problem have been treated by a number of authors, there are still many gaps. In particular, there are no minimax lower bounds for an estimator of $\Phi$ for an arbitrary function $\varphi$. The general upper risk bounds cover only the case when the Fourier transform of $\varphi$ exists. Moreover, no theory exists for estimating $\Phi$ when vector of observations is sparse. In addition, until now, the related problem of estimation of functionals $\Phi_n = n^{-1} \sum_{i=1}^n \varphi(\theta_i)$ in indirect observations have been treated as a separate problem with no connection to estimation of $\Phi$. The objective of the present paper is to fill in the gaps and develop the general minimax theory of estimation of $\Phi$ and $\Phi_n$. We offer a general approach to estimation of $\Phi$ (and $\Phi_n$) and provide the upper and the minimax lower risk bounds in the case when function $\varphi$ is square integrable. Furthermore, we extend the theory to the case when Fourier transform of $\varphi$ does not exist and $\Phi$ can be presented as a linear functional of the Fourier transform of $f$ and its derivatives. Finally, we generalize our results to handle the situation when vector $\theta$ is sparse. As a direct application of the proposed theory, we obtain multiple new results and automatically recover existing ones for a variety of problems such as estimation of the $(2M+1)$-th absolute moment or a generalized moment of the deconvolution density, estimation of the mixing cdf or estimation of the mixing pdf with classical and Berkson errors.