一类带误差模型密度函数导数的小波最优估计

项目名称： 一类带误差模型密度函数导数的小波最优估计

项目编号： No.11271038

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 刘有明

作者单位： 北京工业大学

项目金额： 50万元

中文摘要： 在统计学和计量经济学中，带误差统计模型密度函数及导数估计具有重要的理论意义和应用价值。传统的反卷积核方法存在两个缺陷：一是对于某些密度函数，带宽的选择复杂；二是作为一种线性估计，在许多情形无法给出最优收敛阶.小波方法弥补了这些局限. Lounici, Fan, Nickl 等人利用正交小波基在Besov 空间中研究了带误差密度函数的估计，并取得了重要成果。 由于未见小波方法研究带误差模型密度函数导函数的估计, 本项目拟在Besov 空间中研究一类带误差模型密度函数导数的小波估计，以及最优收敛阶。为此，我们首先讨论密度函数本身的风险估计，以完善Fan, Nickl等人的工作；其次利用非标准型方法研究密度函数导数的小波估计；最后尝试将所得结果推广到高维情形。

中文关键词： 密度函数；L_p 风险；最优性；小波估计；误差模型

英文摘要： In statistics and econometrics, the estimation of density and its derivative for a statistical model with errors plays important roles in both theory and applications. The traditional deconvoluting kernel method has two disadvantages: One is the complexity of bandwidth choice for some densities; Another is that as a linear estimation, it can not give optimal convergence rates in many cases. The wavelet method avoids these shortcomings. The wavelet estimations of densities with errors have been studied by Lounici, Fan, Nickl and etc. They receive excellent results in Besov spaces by using orthogonal wavelets. We try to study wavelet estimations of density derivatives for a class of models with errors in Besov spaces, as well as the optimal convergence rates, because there is not any discussion in this direction: First, estimations for densities will be investigated, in order to extend Fan and Nickl's work; Second, we study wavelet optimal estimations for density derivatives; Finally, it will be considered that the received results are generalized to high dimensional cases.

英文关键词： density function；L_p risk；optimality；wavelet estimation；model with error