Statistical procedures for analyzing data that are in the form of curves and of infinite dimension are provided by functional data analysis. Functional principal component analysis is widely used in the study of functional data, since it allows finite-dimensional analysis of a problem that is intrinsically infinite dimensional. In this article, when considering smoothed functional principal component analysis (SFPCA), we first briefly review Silverman’s method for SFPCA. Then we give a modification of the Silverman’s method for SFPCA and investigate the performance of the modification through stochastic expansions. The modification is based on considering another parameter with the smoothing parameter proposed by Silverman (1996). We study the consistency under suitable conditions theoretically and show that adding the new parameter partly improves the performance of the eigenfunctions estimators toward having smaller error. We also show this improvement through a simulation study, numerically.
Eigenfunction Eigenvalue Functional data analysis Smoothed functional principal component analysis Stochastic expansion
AMS Subject Classification
62H25 62M99 60H25
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Dauxois, J., A. Pousse, and Y. Romain. 1982. Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference. J. Multivariate Anal., 12, 136–154.MathSciNetCrossRefGoogle Scholar
Hall, P., and M. Hosseini-Nasab. 2006. On properties of functional principal components analysis. J. R. Stat. Soc. Ser. B Stat. Methodol. 68, 109–126.MathSciNetCrossRefGoogle Scholar
Hall, P., and M. Hosseini-Nasab. 2009. Theory for high-order bounds in functional principal components analysis. Math. Proc. Camb. Philos. Soc., 149, 225–56.MathSciNetCrossRefGoogle Scholar
He, G. Z., H.-G. Muller, and J.-L. Wang. 2003. Functional canonical analysis for square integrable stochastic processes. J. Multivar. Anal., 85, 54–77.MathSciNetCrossRefGoogle Scholar
Hoerl, A. E., and R. W. Kennard. 1970a. Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12, 55–67.CrossRefGoogle Scholar
Hoerl, A. E., and R. W. Kennard. 1970b. Ridge regression: Application for nonorthogonal problems. Technometrics, 12, 69–82.CrossRefGoogle Scholar