Open Access
July 2007 Statistical inferences for functional data
Jin-Ting Zhang, Jianwei Chen
Ann. Statist. 35(3): 1052-1079 (July 2007). DOI: 10.1214/009053606000001505


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. However, 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.


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Jin-Ting Zhang. Jianwei Chen. "Statistical inferences for functional data." Ann. Statist. 35 (3) 1052 - 1079, July 2007.


Published: July 2007
First available in Project Euclid: 24 July 2007

zbMATH: 1129.62029
MathSciNet: MR2341698
Digital Object Identifier: 10.1214/009053606000001505

Primary: 62G07
Secondary: 62G10 , 62J12

Keywords: Asymptotic Gaussian process , Asymptotic normal distribution , functional data , hypothesis test , local polynomial smoothing , nonparametric estimation , reconstructed individual functions , root-n consistent

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 3 • July 2007
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