一类带误差模型密度函数导数的小波最优估计

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.

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。

密度估计在统计学和计量经济学中具有重要的理论意义和应用价值。针对某些密度函数, 传统的核方法在带宽选择及最优估计方面存在缺陷, 小波估计可以弥补这一局限.. 本项目在Besov空间中研究带误差密度函数及导数的小波估计, 取得主要成果如下:. 1. 针对加法噪声模型, 给出了L_p风险的最优小波估计, 它是Fan-Koo, Lounici-Nickl等人工作的拓广. 当待估函数超级光滑(supersmooth)时, 最优收敛阶得到了改善, 且补充了Pensky, Comte 等人的工作.. 2. 针对Fourier震荡噪声, 首先研究了密度函数导数的线性小波估计, 推广了Delaigle-Meister定理; 其次讨论了非线性小波估计, 得到自适应性和更好的收敛阶; 最后将独立随机样本放松为负相协(Negatively Associated)情形, 得到L_p风险估计, 其结果优于Chaubey等人的估计.. 3. 针对一类乘法噪声模型及负相协随机样本, 研究了小波估计器的 L_p相合性. 在Besov空间中给出了线性及非线性小波估计器 L_p风险的收敛阶.