使用传播方法的逐点自适应中的参数调整
Parameter tuning in pointwise adaptation using a propagation approach
摘要：
本文讨论了线性反问题中单变量对象如回归函数在给定点的值或线性泛函的自适应估计问题。我们认为一个自适应程序起源于Lepski [理论Probab。35（1990）454- 466.]它以数据驱动的方式从按可变性排序的给定估计类别中选择一个估计。使用此过程和类似过程的一个严重问题是选择某些调优参数，如阈值。数值结果表明，理论上推荐的建议似乎过于保守，导致强烈的过平滑效应。仔细选择过程的参数对于获得合理的估计质量是极其重要的。本文的主要贡献是新的方法来选择参数的过程中，通过提供规定的行为，在简单的参数的情况下，得到的估计。我们建立了一个非渐近的“预言”界，这表明估计的风险是，对数乘数，等于风险的“预言”估计是最佳选择从给定的家庭。数值研究表明，在一些模拟的例子所产生的程序的良好性能。
英文摘要：
This paper discusses the problem of adaptive estimation of a univariate object like the value of a regression function at a given point or a linear functional in a linear inverse problem. We consider an adaptive procedure originated from Lepski [Theory Probab. Appl. 35 (1990) 454--466.] that selects in a data-driven way one estimate out of a given class of estimates ordered by their variability. A serious problem with using this and similar procedures is the choice of some tuning parameters like thresholds. Numerical results show that the theoretically recommended proposals appear to be too conservative and lead to a strong oversmoothing effect. A careful choice of the parameters of the procedure is extremely important for getting the reasonable quality of estimation. The main contribution of this paper is the new approach for choosing the parameters of the procedure by providing the prescribed behavior of the resulting estimate in the simple parametric situation. We establish a non-asymptotical "oracle" bound, which shows that the estimation risk is, up to a logarithmic multiplier, equal to the risk of the "oracle" estimate that is optimally selected from the given family. A numerical study demonstrates a good performance of the resulting procedure in a number of simulated examples.
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