论文标题
功能输出的功能线性模型中的自适应非参数估计
Adaptive nonparametric estimation in the functional linear model with functional output
论文作者
论文摘要
在本文中，我们考虑了一个功能线性回归模型，其中协变量和响应变量都是功能随机变量。我们解决了该模型中条件期望运算符的最佳非参数估计的问题。首先引入了有限维尺寸子空间的投影估计器的集合。在这些子空间由（经验）PCA函数基础产生的情况下，我们为均方根预测误差提供了非反应偏置偏差分解。由于选择了最佳投影维度的模型选择设备，因此实现了自动权衡：惩罚造影剂估算器满足了甲骨文类型的不平等，因此从适应性的角度来看是最佳的。这些上限使我们能够在椭圆形平滑度空间上得出收敛速率。在最小值的意义上，该速率被证明是最佳的：它们与Minimax风险的下限匹配，这也证明了这一点。最后，我们进行了一项数值研究，超过模拟数据和两个真实数据集。
In this paper, we consider a functional linear regression model, where both the covariate and the response variable are functional random variables. We address the problem of optimal nonparametric estimation of the conditional expectation operator in this model. A collection of projection estimators over finite dimensional subspaces is first introduce. We provide a non-asymptotic bias-variance decomposition for the Mean Square Prediction error in the case where these subspaces are generated by the (empirical) PCA functional basis. The automatic trade-off is realized thanks to a model selection device which selects the best projection dimensions: the penalized contrast estimator satisfies an oracle-type inequality and is thus optimal in an adaptive point of view. These upper-bounds allow us to derive convergence rates over ellipsoidal smoothness spaces. The rates are shown to be optimal in the minimax sense: they match with a lower bound of the minimax risk, which is also proved. Finally, we conduct a numerical study, over simulated data and over two real-data sets.
