Meyer小波在反褶积和密度估计中的应用
On the usefulness of Meyer wavelets for deconvolution and density estimation
摘要：
本文的目的是显示实用的迈耶小波的经典问题的密度估计和密度反褶积噪声观测。通过使用这样的小波，经验小波系数的计算依赖于数据的快速傅里叶变换和Meyer小波是带限函数的事实。这使得这样的估计非常简单的计算，这避免了评估小波在非二进点，这是经典的基于小波的密度估计的主要缺点的问题。我们的方法是基于长期的经验小波系数与随机阈值依赖于每个系数的方差估计的阈值。这样的估计，以实现相同的性能的Oracle估计的对数项。这些估计也实现了一个大类的Besov空间的收敛速度近极大极小。仿真研究表明，该估计器对直接密度估计和密度反卷积问题都具有良好的有限样本性能。
英文摘要：
The aim of this paper is to show the usefulness of Meyer wavelets for the classical problem of density estimation and for density deconvolution from noisy observations. By using such wavelets, the computation of the empirical wavelet coefficients relies on the fast Fourier transform of the data and on the fact that Meyer wavelets are band-limited functions. This makes such estimators very simple to compute and this avoids the problem of evaluating wavelets at non-dyadic points which is the main drawback of classical wavelet-based density estimators. Our approach is based on term-by-term thresholding of the empirical wavelet coefficients with random thresholds depending on an estimation of the variance of each coefficient. Such estimators are shown to achieve the same performances of an oracle estimator up to a logarithmic term. These estimators also achieve near-minimax rates of convergence over a large class of Besov spaces. A simulation study is proposed to show the good finite sample performances of the estimator for both problems of direct density estimation and density deconvolution.
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