Adaptive functional linear regression
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自适应函数型线性回归
- 研究问题
- 在标量响应依赖随机函数的线性回归中，如何不依赖未知斜率与协方差特征选择调节参数，并兼顾预测、斜率和导数估计？
- 方法与思路
- 作者结合模型选择与 Lepski 方法，在随机可选参数集合上最小化随机惩罚对比函数，为阈值投影估计选择数据驱动参数。
- 材料陈述的贡献
- 摘要称该数据驱动程序在多类斜率函数和协方差算子上达到常数因子意义下的极小极大收敛率，并给出模拟展示。
- 阅读关注点
- 阅读全文时应核对函数类与协方差算子条件、惩罚项和可选集合的构造，以及不同损失下的收敛率与调参计算要求。
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原文摘要
We consider the estimation of the slope function in functional linear regression, where scalar responses are modeled in dependence of random functions. Cardot and Johannes [J. Multivariate Anal. 101 (2010) 395–408] have shown that a thresholded projection estimator can attain up to a constant minimax-rates of convergence in a general framework which allows us to cover the prediction problem with respect to the mean squared prediction error as well as the estimation of the slope function and its derivatives. This estimation procedure, however, requires an optimal choice of a tuning parameter with regard to certain characteristics of the slope function and the covariance operator associated with the functional regressor. As this information is usually inaccessible in practice, we investigate a fully data-driven choice of the tuning parameter which combines model selection and Lepski’s method. It is inspired by the recent work of Goldenshluger and Lepski [Ann. Statist. 39 (2011) 1608–1632]. The tuning parameter is selected as minimizer of a stochastic penalized contrast function imitating Lepski’s method among a random collection of admissible values. This choice of the tuning parameter depends only on the data and we show that within the general framework the resulting data-driven thresholded projection estimator can attain minimax-rates up to a constant over a variety of classes of slope functions and covariance operators. The results are illustrated considering different configurations which cover in particular the prediction problem as well as the estimation of the slope and its derivatives. A simulation study shows the reasonable performance of the fully data-driven estimation procedure.