Prediction in functional linear regression
Prediction in functional linear regression
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DOI：
10.1214/009053606000000830
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发表时间：
2006-10-01
影响因子：
4.5
通讯作者：
Hall, Peter
中科院分区：
文献类型：
--
作者：
Cai, T. Tony;Hall, Peter
在函数数据分析的线性回归中，对斜率函数的估计方法进行了大量的研究。然而，在更传统的有限维回归的情况下，对斜率的实际兴趣主要集中在其预测目的的应用上，而不是其本身的重要性。我们表明，问题的斜率函数估计，并预测从估计的斜率函数，有非常不同的特点。虽然前者本质上是非参数的，但后者可以是非参数的或半参数的。特别地，如果预测变量是一个充分光滑的函数，则预测变量的最优均方收敛速度是n（-1），其中n表示样本大小。在其他情况下，收敛以严格慢于n（-1）的多项式速率发生。在这两个区域之间的边界处，均方收敛率仅比n（-1）小一个对数因子。更一般地说，给定解释变量的特定值，回归模型中平均响应的预测值的收敛速率由预测函数、模型中斜率函数和解释变量分布的自协方差函数的平滑度之间的微妙相互作用决定。
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 semi-parametric. 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.