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Title: Statistical inferences for functional data
Authors: Zhang, J.-T. 
Chen, J.
Keywords: Asymptotic Gaussian process
Asymptotic normal distribution
Functional data
Hypothesis test
Local polynomial smoothing
Nonparametric estimation
Reconstructed individual functions
Root-n consistent
Issue Date: Jul-2007
Citation: Zhang, J.-T., Chen, J. (2007-07). Statistical inferences for functional data. Annals of Statistics 35 (3) : 1052-1079. ScholarBank@NUS Repository.
Abstract: With modern technology development, functional data are being observed frequently in many scientific fields. A popular method for analyzing such functional data is "smoothing first, then estimation." That is, statistical inference such as estimation and hypothesis testing about functional data is conducted based on the substitution of the underlying individual functions by their reconstructions obtained by one smoothing technique or another. How-ever, little is known about this substitution effect on functional data analysis. In this paper this problem is investigated when the local polynomial kernel (LPK) smoothing technique is used for individual function reconstructions. We find that under some mild conditions, the substitution effect can be ignored asymptotically. Based on this, we construct LPK reconstruction-based estimators for the mean, covariance and noise variance functions of a functional data set and derive their asymptotics. We also propose a GCV rule for selecting good bandwidths for the LPK reconstructions. When the mean function also depends on some time-independent covariates, we consider a functional linear model where the mean function is linearly related to the covariates but the covariate effects are functions of time. The LPK reconstruction-based estimators for the covariate effects and the covariance function are also constructed and their asymptotics are derived. Moreover, we propose a L2-norm-based global test statistic for a general hypothesis testing problem about the covariate effects and derive its asymptotic random expression. The effect of the bandwidths selected by the proposed GCV rule on the accuracy of the LPK reconstructions and the mean function estimator is investigated via a simulation study. The proposed methodologies are illustrated via an application to a real functional data set collected in climatology. © Institute of Mathematical Statistics, 2007.
Source Title: Annals of Statistics
ISSN: 00905364
DOI: 10.1214/009053606000001505
Appears in Collections:Staff Publications

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