2006/10/01 by T. Tony Cai, Tommaso Cai, Peter Hall · 3 citations
Mathematics · #Advanced Statistical Methods and Models #Statistical Methods and Bayesian Inference #Statistical Methods and Inference #math.ST #msc:62G20 #msc:62J05 #stat.TH
paper · pdf · doi:10.1214/009053606000000830
published as Annals of Statistics 2006, Vol. 34, No. 5, 2159-2179 · Published at http://dx.doi.org/10.1214/009053606000000830 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2006/10/01 · arxiv created 2007/02/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
There has been substantial recent work on methods for estimating the slope function in linear regression for functional data analysis. However, as in the case of more conventional finite-dimensional regression, much of the practical interest in the slope centers on its application for the purpose of prediction, rather than on its significance in its own right. We show that the problems of slope-function estimation, and of prediction from an estimator of the slope function, have very different characteristics. While the former is intrinsically nonparametric, the latter can be either nonparametric or semiparametric. In particular, the optimal mean-square convergence rate of predictors is n−1, where n denotes sample size, if the predictand is a sufficiently smooth function. In other cases, convergence occurs at a polynomial rate that is strictly slower than n−1. At the boundary between these two regimes, the mean-square convergence rate is less than n−1 by only a logarithmic factor. More generally, the rate of convergence of the predicted value of the mean response in the regression model, given a particular value of the explanatory variable, is determined by a subtle interaction among the smoothness of the predictand, of the slope function in the model, and of the autocovariance function for the distribution of explanatory variables.