基于广义反卷积模型多元密度函数的小波最优估计

The generalized deconvolution density estimate model is an important extension of traditional deconvolution one, and it is widely used in many engineering fields such as statistics and financial mathematics. The classical deconvolution density estimations assume that the presence of random noise is a deterministic event, while the observation data do not necessarily contain the errors in practical applications. The real density function is obtained from some real data and some noise-polluted data, which is the research content of the generalized deconvolution density estimation model..Although the classical kernel density estimation has made important progress, its bandwidth selection rule is complex and the function spaces are limited. This project aims to study the optimal generalized deconvolution estimations for multivariate density functions over anisotropic Besov spaces by using wavelets, based on the time-frequency localization characteristics and the function of describing function spaces of wavelet bases. First, combining with noise information, we construct wavelet anisotropic density estimators and show their convergence rates over global risk for moderately and severely ill-posed noises respectively. Second, we consider lower bounds of all possible estimators for generalized deconvolution density estimations under two types of noises. Finally, a data-driven version is provided for adaptivity and its convergence performance is investigated; We try to reduce the influence of "the curse of dimensionality" by using the independent structure of the estimated density.

广义反卷积密度估计是传统反卷积密度估计的重要推广，且在统计学、金融数学等众多工程领域中具有广泛应用。传统的反卷积密度估计假定随机噪声的存在是确定性事件，而在实际应用中观测数据不一定全部含有误差。从部分真实数据和部分被噪声污染的数据中得到真实数据的分布，即为广义反卷积密度估计模型的研究内容。.经典的核密度估计虽然取得重要进展，但其带宽选择复杂且函数空间受到局限。本项目利用小波基的时频局部化特性及刻画函数空间的功能，在各向异性的Besov空间中研究广义反卷积多元密度函数的小波最优估计。首先，结合噪声信息构造各向异性密度函数的小波估计器，针对适度病态及严重病态噪声分别研究其整体风险误差；其次，讨论该模型在两类噪声条件下密度函数所有估计器的下界，进一步分析所构造估计器的最优性；最后，将上述估计器改进为基于数据驱动的自适应估计器，并研究其收敛阶；尝试利用待估函数的独立结构降低“维数灾难”的影响。

广义反卷积密度估计是传统反卷积密度估计的重要推广，且在统计学、金融数学等众多工程领域中具有广泛应用。从部分真实数据和部分被噪声污染的数据中得到真实数据的分布，即为广义反卷积密度估计模型的研究内容。.本项目基于广义反卷积模型研究了多元密度函数的自适应小波最优估计。具体地，首先在Besov空间中利用小波方法建立了多元密度函数的广义反卷积估计，讨论了小波估计器的Lp极大风险；其次在局部Hölder空间中，基于广义反卷积模型在p-阶点态风险下研究了自适应小波最优(或近似最优)估计；最后利用数据驱动技术给出了完全自适应的小波估计，进一步利用待估函数的独立结构降低了维数灾难的影响。