Donoho and Johnstone (1994) proposed a method for reconstructing an unknown function f on [0,1] from noisy data d/sub i/=f(t/sub i/)+/spl sigma/z/sub i/, i=0, ..., n-1,t/sub i/=i, where the z/sub i/ are independent and identically distributed standard Gaussian random variables. The reconstruction f/spl circ/*/sub n/ is defined in the wavelet domain by translating all the empirical wavelet coefficients of d toward 0 by an amount /spl sigma//spl middot//spl radic/(2log (n)). The authors prove two results about this type of estimator. [Smooth]: with high probability f/spl circ/*/sub n/ is at least as smooth as f, in any of a wide variety of smoothness measures. [Adapt]: the estimator comes nearly as close in mean square to f as any measurable estimator can come, uniformly over balls in each of two broad scales of smoothness classes. These two properties are unprecedented in several ways. The present proof of these results develops new facts about abstract statistical inference and its connection with an optimal recovery model.
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机译：Donoho和Johnstone（1994）提出了一种从噪声数据d / sub i / = f（t / sub i /）+ / spl sigma / z / sub i /，i重建[0,1]上未知函数f的方法= 0，...，n-1，t / sub i / = i / n，其中z / sub i /是独立且均布的标准高斯随机变量。重构f / spl circ / * / sub n /是在小波域中定义的，方法是将所有d的经验小波系数向0转换量/ spl sigma // spl middot // spl radic /（2log（n）/ n）。作者证明了有关这种估计量的两个结果。 [平滑]：在多种平滑度度量中的任何一种下，f / spl circ / * / sub n /的可能性至少与f一样平滑。 [适应]：估计器的均方差几乎与任何可测量的估计器所能达到的f相等，均匀地分布在两个广泛的平滑度等级中的每个球上。这两种特性在几种方面都是空前的。这些结果的当前证明开发了有关抽象统计推断及其与最佳恢复模型的联系的新事实。
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