UNIFORM CONVERGENCE RATES FOR NONPARAMETRIC REGRESSION AND PRINCIPAL COMPONENT ANALYSIS IN FUNCTIONAL/LONGITUDINAL DATA
UNIFORM CONVERGENCE RATES FOR NONPARAMETRIC REGRESSION AND PRINCIPAL COMPONENT ANALYSIS IN FUNCTIONAL/LONGITUDINAL DATA
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DOI：
10.1214/10-aos813
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发表时间：
2010-12-01
影响因子：
4.5
通讯作者：
Hsing, Tailen
中科院分区：
文献类型：
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作者：
Li, Yehua;Hsing, Tailen
我们考虑功能/纵向数据的均值和协方差函数的非参数估计。强一致收敛速度的估计是局部线性平滑。我们的结果是在一个统一的框架中得到的，在这个框架中，每个曲线/聚类内的观测值的数量可以是相对于样本大小的任何速率。我们表明，该程序的收敛速度取决于样本曲线的数量和每个曲线上的观测数。对于稀疏函数数据，这些速率等价于非参数回归中的最优速率。对于密集的函数数据，可以通过适当选择带宽来实现根-n收敛率。我们进一步推导出几乎肯定的主成分分析的收敛速度，使用估计的协方差函数。仿真结果表明，与模拟研究。
We consider nonparametric estimation of the mean and covariance functions for functional/longitudinal data. Strong uniform convergence rates are developed for estimators that are local-linear smoothers. Our results are obtained in a unified framework in which the number of observations within each curve/cluster can be of any rate relative to the sample size. We show that the convergence rates for the procedures depend on both the number of sample curves and the number of observations on each curve. For sparse functional data, these rates are equivalent to the optimal rates in nonparametric regression. For dense functional data, root-n rates of convergence can be achieved with proper choices of bandwidths. We further derive almost sure rates of convergence for principal component analysis using the estimated covariance function. The results are illustrated with simulation studies.