Consistent Nonparametric Regression for Functional Data Under the Stone–Besicovitch Conditions
Consistent Nonparametric Regression for Functional Data Under the Stone–Besicovitch Conditions
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Stone-Besicovitch 条件下函数数据的一致非参数回归
DOI：
10.1109/tit.2012.2209628
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发表时间：
2012
影响因子：
2.5
通讯作者：
Pamela Llop
中科院分区：
文献类型：
--
作者：
L. Forzani;R. Fraiman;Pamela Llop
本文讨论了无限维非参数回归估计问题。我们首先将Stone的开创性结果推广到度量空间的情况，当解释变量的概率度量是紧的时。然后，在假设条件稍有变化的情况下，给出并证明了一般度量度量空间的定理。根据这一结果，当回归函数有界且Besicovitch条件成立时，我们得到了k-NN和核估计的均方相合性。我们还证明了，对于一致核估计，Besicovitch条件也是对几乎每个x达到L1相合性的必要条件。
In this paper, we address the problem of nonparametric regression estimation in the infinite-dimensional setting. We start by extending the Stone's seminal result to the case of metric spaces when the probability measure of the explanatory variables is tight. Then, under slight variations on the hypotheses, we state and prove the theorem for general metric measure spaces. From this result, we derive the mean square consistency of the k-NN and kernel estimators if the regression function is bounded and the Besicovitch condition holds. We also prove that, for the uniform kernel estimate, the Besicovitch condition is also necessary in order to attain L1 consistency for almost every x.