Optimal pointwise adaptive methods in nonparametric estimation
Optimal pointwise adaptive methods in nonparametric estimation
复制标题
非参数估计中的最优逐点自适应方法
DOI：
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
V. Spokoiny
中科院分区：
文献类型：
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作者：
O. Lepski;V. Spokoiny
本文研究了在给定点上由噪声数据对函数进行最优自适应估计的问题。两个程序被证明是渐近最优的不同的设置。首先研究了在给定核条件下非参数逐点核估计的带宽选择问题。我们提出了一个带宽选择过程，并证明其渐近意义下的最优性。此外，这种最优性不仅在具有可变带宽的核估计中得到陈述。所得估计量在所有可行估计量中是渐近最优的。这个过程的重要特点是，它是完全自适应的，它适用于一个非常广泛的一类函数服从温和的正则性限制。利用它，可达到的估计精度取决于函数本身，并表示为与该函数和给定核相对应的理想自适应带宽。第二个过程可以被认为是第一个过程的特殊化，在定性假设下，待估计的函数属于某个保持器类（β，L），未知参数β，L。这个假设允许我们以最优的方式选择一个核族，并且由此产生的过程在β ≤ 2的任何自适应范围内的自适应意义上似乎是渐近最优的。
The problem of optimal adaptive estimation of a function at a given point from noisy data is considered. Two procedures are proved to be asymptotically optimal for different settings. First we study the problem of bandwidth selection for nonparametric pointwise kernel estimation with a given kernel. We propose a bandwidth selection procedure and prove its optimality in the asymptotic sense. Moreover, this optimality is stated not only among kernel estimators with a variable bandwidth. The resulting estimator is asymptotically optimal among all feasible estimators. The important feature of this procedure is that it is fully adaptive and it works for a very wide class of functions obeying a mild regularity restriction. With it the attainable accuracy of estimation depends on the function itself and is expressed in terms of the ideal adaptive bandwidth corresponding to this function and a given kernel. The second procedure can be considered as a specialization of the first one under the qualitative assumption that the function to be estimated belongs to some Holder class Σ(β, L) with unknown parameters β, L. This assumption allows us to choose a family of kernels in an optimal way and the resulting procedure appears to be asymptotically optimal in the adaptive sense in any range of adaptation with β ≤ 2.