MULTIVARIATE WAVELET THRESHOLDING IN ANISOTROPIC FUNCTION SPACES
MULTIVARIATE WAVELET THRESHOLDING IN ANISOTROPIC FUNCTION SPACES
复制标题
各向异性函数空间中的多元小波阈值处理
DOI：
--
复制
发表时间：
2000
期刊：
影响因子：
--
通讯作者：
Michael H. Neumann
中科院分区：
文献类型：
--
作者：
Michael H. Neumann
众所周知，标准（各向同性）光滑条件下的多元曲线估计遭受“维数灾难”。这反映在标准渐近设置中严重恶化的收敛速率上。如果要估计的曲线在不同方向上具有不同的平滑特性，并且估计方案能够在某些方向上利用较低的复杂度，则可能有比对应于各向同性平滑先验的收敛速率更好的收敛速率。我们考虑典型的情况下，各向异性的平滑类，并探讨如何适当的小波估计可以利用这种限制的曲线，需要适应不同的光滑性在不同的方向。事实证明，非线性阈值与各向异性的多元小波基下，各向异性类型的平滑先验的收敛速度最佳。我们推导出“信号加高斯白色噪声”模型的渐近结果，其中随着样本量的增加，降低的噪声水平模仿标准渐近性。
It is well known that multivariate curve estimation under standard (isotropic) smoothness conditions suffers from the "curse of dimensionality". This is reflected by rates of convergence that deteriorate seriously in standard asymp- totic settings. Better rates of convergence than those corresponding to isotropic smoothness priors are possible if the curve to be estimated has different smoothness properties in different directions and the estimation scheme is capable of making use of a lower complexity in some of the directions. We consider typical cases of anisotropic smoothness classes and explore how appropriate wavelet estimators can exploit such restrictions on the curve that require an adaptation to different smooth- ness properties in different directions. It turns out that nonlinear thresholding with an anisotropic multivariate wavelet basis leads to optimal rates of convergence un- der smoothness priors of anisotropic type. We derive asymptotic results in the model "signal plus Gaussian white noise", where a decreasing noise level mimics the standard asymptotics with increasing sample size.