Mathematics – Statistics Theory
Scientific paper
2007-02-22
Annals of Statistics 2006, Vol. 34, No. 5, 2159-2179
Mathematics
Statistics Theory
Published at http://dx.doi.org/10.1214/009053606000000830 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst
Scientific paper
10.1214/009053606000000830
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.
Cai Tony T.
Hall Peter
No associations
LandOfFree
Prediction in functional linear regression does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Prediction in functional linear regression, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Prediction in functional linear regression will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-102257