A simple method to construct confidence bands in functional linear regression
A simple method to construct confidence bands in functional linear regression
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DOI：
10.5705/ss.202017.0208
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发表时间：
2016-12
影响因子：
1.4
通讯作者：
M. Imaizumi;Kengo Kato
中科院分区：
文献类型：
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作者：
M. Imaizumi;Kengo Kato
本文提出了一种简单的方法来构建置信带，在主成分分析（PCA）为基础的估计，为斜率函数的功能线性回归模型与标量响应变量和功能预测变量。基于PCA的估计量是具有估计的基函数的序列估计量，因此构造其有效置信带是一个非平凡的挑战。我们提出了一个置信带，其目的是在覆盖的斜率函数在“最”的点与预先指定的概率（水平），并证明其渐近有效性在适当的正则性条件下。重要的是，这是第一篇论文，推导出的PCA为基础的估计具有理论依据的置信带。我们还提出了一种实用的方法来选择基于PCA的估计中使用的截止水平，并进行数值研究，以验证所提出的置信带的有限样本性能。最后，我们将我们的方法应用于光谱数据，并讨论了我们的方法的扩展，其中存在额外的向量值回归的情况。
This paper develops a simple method to construct confidence bands, centered at a principal component analysis (PCA) based estimator, for the slope function in a functional linear regression model with a scalar response variable and a functional predictor variable. The PCA-based estimator is a series estimator with estimated basis functions, and so construction of valid confidence bands for it is a non-trivial challenge. We propose a confidence band that aims at covering the slope function at "most" of points with a prespecified probability (level), and prove its asymptotic validity under suitable regularity conditions. Importantly, this is the first paper that derives confidence bands having theoretical justifications for the PCA-based estimator. We also propose a practical method to choose the cut-off level used in PCA-based estimation, and conduct numerical studies to verify the finite sample performance of the proposed confidence band. Finally, we apply our methodology to spectrometric data, and discuss extensions of our methodology to cases where additional vector-valued regressors are present.