非紧支非正则多元密度函数的自适应小波估计

One important application of Wavelet Analysis is nonparametric density estimation. Donoho and many other researchers always assume that the estimated functions have compact support and some regularity, when they study the adaptive and optimal wavelet estimations of p risk for density estimators. If the density functions are non-compact and irregular, Juditsky, Reynaud-Bouret and ect provide nice solution to the above problem for some special cases.. This project studies the adaptive and optimal (or sub-optimal) estimations of p risk for non-compact and irregular density functions in Besov spaces. Firstly, we try to solve that problem for isotropic density functions in Besov space, motivated by the work of Lepski and Reynaud-Bouret; Secondly, the same problem will be considered for multivariate density functions under some independent structure. It is hoped to make up a disadvantage (the complicated bandwidth selection) of kernel estimation and to keep the advantage avoiding curse of dimensionality; Lastly, we discuss the adaptive and optimal wavelet estimations of p risk for densities with moderately and severely ill-posed noises, which generalizes Comte-Lacour’s 2 risk estimation.

小波分析的一个重要应用是非参数密度估计。关于密度函数p阶风险的自适应小波最优估计问题研究，Donoho及其后来的大多数学者均假定待估函数具有紧支集与某种正则性。针对非紧支非正则密度函数，Juditsky与Reynaud-Bouret等人在一些特殊情形给出了上述问题的理想解。.本项目拟在Besov空间中研究非紧支非正则密度函数p阶风险的自适应小波最优（或次优）估计。首先借鉴Lepski与Reynaud-Bouret等人的工作，讨论各向同性密度函数的自适应小波最优估计；其次利用小波方法研究具有某种独立结构的多元密度函数的同一问题，以弥补核估计带宽选择过于复杂的缺陷，同时保留核方法避免维数灾难的优点；最后给出带适度病态噪声及严重病态噪声密度函数p阶风险的自适应小波估计，期望推广Comte-Lacour的2阶风险估计。

利用小波方法估计密度函数已取得了大量丰硕的成果，但一般均假定待估函数具有紧支集。Reynaud-Bouret等人在无紧支条件下给出了$L^2$风险的最优估计，并提出$L^p$风险的最优估计问题。. 本项目在$p\geq 2$时部分地正面回答了上述问题。为研究多元密度函数的点态风险估计，我们引入了局部异性的Holder空间，并在其上给出了数据驱动的自适应小波次优估计；和经典的核估计相比，它弥补了带宽选择复杂的缺陷，同时保留了可避免维数灾难的优势。. 最后研究了一类广义反卷积多元密度函数在点态风险意义下的次优估计问题。所得结果包含若干已有工作作为特例。此外还证明了为实现自适应性，损失$ln n$因子的收敛阶是无法避免的。