Single and multiple index functional regression models with nonparametric link
Single and multiple index functional regression models with nonparametric link
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DOI：
10.1214/11-aos882
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发表时间：
2011-06
影响因子：
4.5
通讯作者：
Dong Chen;P. Hall;H. Muller
中科院分区：
文献类型：
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作者：
Dong Chen;P. Hall;H. Muller
从统计学的角度来看，用于函数数据回归的完全非参数方法的准确性较差，这反映了它们的收敛速度慢于高维函数估计的非参数速度。这种困难导致强调所谓的功能线性模型，这是更灵活的比常见的线性模型在有限的维度，但仍然施加结构约束的预测和响应之间的关系。最近的进展已经扩展了线性的方法，使用它与链接功能，并考虑多个指标，但这种技术的灵活性仍然是有限的。例如，链路可以参数化建模或仅在网格上建模，或者可以受到诸如单调性之类的假设的约束;多个索引已经通过进行有限维假设来建模。在本文中，我们介绍了一种新的技术，估计非参数化的链接函数，我们提出了一种方法，多指标建模使用自适应定义的线性预测的功能数据。我们表明，我们的方法能够预测多项式收敛速度。我们的方法的有限样本性能的模拟研究，并说明了一个应用程序的功能回归问题。
Fully nonparametric methods for regression from functional data have poor accuracy from a statistical viewpoint, reflecting the fact that their convergence rates are slower than nonparametric rates for the estimation of high-dimensional functions. This difficulty has led to an emphasis on the so-called functional linear model, which is much more flexible than common linear models in finite dimension, but nevertheless imposes structural constraints on the relationship between predictors and responses. Recent advances have extended the linear approach by using it in conjunction with link functions, and by considering multiple indices, but the flexibility of this technique is still limited. For example, the link may be modeled parametrically or on a grid only, or may be constrained by an assumption such as monotonicity; multiple indices have been modeled by making finite-dimensional assumptions. In this paper we introduce a new technique for estimating the link function nonparametrically, and we suggest an approach to multi-index modeling using adaptively defined linear projections of functional data. We show that our methods enable prediction with polynomial convergence rates. The finite sample performance of our methods is studied in simulations, and is illustrated by an application to a functional regression problem.