小波阈值估计的收敛性及密度函数估计问题的研究

The wavelet thresholding method, proposed by Donoho and etc, has been applied greatly in scientific areas, including filtering, compression denoising, etc. Nowadays, the wavelet thresholding estimators, mainly constructed by orthogonal wavelet, are successfully applied to study the convergence problem of the non-standard form of operators'(e.g. differential operator) wavelet thresholding estimators under the Besov spaces on the whole space(e.g. the real line). Moreover, wavelet methods are also applied successfully to the study of density estimation of nonparametric statistics. This is mainly because wavelets have important properties such as the multiresolution decomposition, characterizations for functional spaces and providing fast algorithm.. Based on these observations, we shall study the following problems: Because non-orthogonal wavelets (e.g. frame) are more flexible, firstly, relaxing the conditions of wavelets to non-orthogonal cases, one will study the convergence of these thresholding estimators, including strengthening restrictions of Besov spaces to bounded domains, to analyze the estimators' convergence problem; Secondly, the estimation of density in statistics, based on non-orthogonal wavelets (frame) and the above results, will be considered and also the revelent problems. Finally, since it is unknown in practice for smoothness of density and sample is lack of independence or is contaminated by noise (error), we will study the smoothness density problem by using nonindependent sample or sample with noise (error)...

小波阈值方法，首先由Donoho等人提出，被广泛应用于滤波、压缩去噪等领域. 现有的小波阈值估计主要是基于正交小波进行的，并被成功地运用到基于R(或n维实空间)上Besov空间的算子(如微分算子)非标准型的小波阈值估计量收敛性问题的研究. 此外，小波还被成功应用于非参数统计中的密度函数估计问题研究，这主要是由于小波具有多尺度分解、刻画函数空间及提供快速算法等重要性质的原因.. 因此本项目拟研究如下问题：由于非正交小波(如标架)有更多的自由度，首先放松限制到非正交情形，研究此时估计量的收敛性问题，包括限制到有界区域上Besov空间的阈值估计量的收敛性；其次研究非正交小波(标架)在统计中基于上述结果的密度估计及相关问题；最后因实际应用中密度光滑性未知及样本不独立或带有噪音(误差)，本项目拟研究基于不独立或含噪音(误差)样本的密度光滑性参数估计问题.

小波阈值方法，首先由Donoho等人提出，被广泛应用于滤波、压缩去噪等领域. 现有的小波阈值估计主要是基于正交小波进行的，并被成功地运用到基于R(或n维实空间)上Besov空间的算子(如微分算子)非标准型的小波阈值估计量收敛性问题的研究. 此外，小波还被成功应用于非参数统计中的密度函数估计问题研究，这主要是由于小波具有多尺度分解、刻画函数空间及提供快速算法等重要性质的原因..本项目对如下问题进行了研究：利用双正交B-样条小波, 我们研究了相关向量Besov空间小波展开的收敛性，特别地加入了散度和非散度自由小波并进行了研究；给出了一个Parserval标架可以伸缩为可分的Hilbert空间的标准正交基的条件，而这对导出一个待检序列是另一个标架的R-对偶的条件是有帮助的；利用张量积正交小波构造的n维小波, 我们研究了相关高维Besov空间下小波阈值估计量的收敛性；最后，在有一些噪声的条件下研究了密度光滑性指标的非参数估计问题。很好的完成了预定的问题。