Spring Seminar Series - May 5, 2005
University of Minnesota
School of Statistics
College of Liberal Arts
Prediction in Functional Linear Regression
T. Tony Cai
Department of Statistics
The Wharton School
University of Pennsylvania
Thursday, May 5, 2005
3:30 PM, 115
Ford Hall
Minneapolis, East Bank Campus
Social at 3:00 PM, 300 Ford Hall
Abstract
There
has been substantial recent work on methods for estimating the
slope function in functional linear model. However, much of the
practical interest in the slope lies 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 an algebraic
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.
This is joint work with Peter Hall.