高维乘法噪声密度的最优小波估计

Density estimation for size-biased data plays important roles in practical applications, because there exists the bias between observed data and real data which can't be directly observed in many cases. Wavelet density estimation has made great achievements in one dimensional space due to the advantages of wavelet basis. In practical problems, More and more study results are affected by many factors, which lead to the problem of multivariate density estimation.. This project aims to study wavelet density estimation in high dimensional Besov spaces. Based on density estimation for biased data, we define a wavelet estimator for multivariate density functions. Firstly, its consistency and convergence rate are investigated in high dimensional Besov space. Secondly, we give a nonlinear thresholding wavelet estimator and compare its risk upper bounds with other estimators' risk lower bounds in high dimensional Besov space. It is hoped that wavelet estimations are optimal.

带乘法噪声密度估计模型在实际应用中具有重要的意义。人们通常不能直接观测到真实数据，而观测到的数据与真实数据之间往往存在着乘法噪声的关系。由于小波基的巨大优势，一维密度函数的小波估计已经取得了丰硕成果。在实际问题中，越来越多的实验结果受到多种因素影响，这就引出了多元密度函数估计的问题。.本项目拟研究小波方法在高维Besov空间密度函数估计方面的应用。基于乘法噪声密度估计模型，构造多元密度函数的小波估计器。首先，证明该估计器的相合性并研究其在高维Besov空间中风险的收敛阶。其次，利用阈值方法构造非线性小波估计器，在高维Besov空间中讨论该估计器的风险上界和任一估计器的风险下界，表明小波估计器的最优性。