CLT in functional linear regression models

• 成果类型: Article
• 署名作者: Cardot, Herve; Mas, Andre; Sarda, Pascal
• 署名单位: Universite de Montpellier; Institut Agro; AgroSup Dijon; Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Universite de Toulouse; Universite de Toulouse - Jean Jaures
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-006-0025-2
• 发表日期: 2007
• 页码: 325-361
• 关键词: statistical view

摘要

We propose in this work to derive a CLT in the functional linear regression model. The main difficulty is due to the fact that estimation of the functional parameter leads to a kind of ill-posed inverse problem. We consider estimators that belong to a large class of regularizing methods and we first show that, contrary to the multivariate case, it is not possible to state a CLT in the topology of the considered functional space. However, we show that we can get a CLT for the weak topology under mild hypotheses and in particular without assuming any strong assumptions on the decay of the eigenvalues of the covariance operator. Rates of convergence depend on the smoothness of the functional coefficient and on the point in which the prediction is made.