Convergence rate of principal component analysis with local-linear smoother for functional data under a unified weighing scheme

ABSTRACT

The unified weighing scheme for the local-linear smoother in analysing functional data can deal with data that are dense, sparse or of neither type. In this paper, we focus on the convergence rate of functional principal component analysis using this method. Almost sure asymptotic consistency and rates of convergence for the estimators of eigenvalues and eigenfunctions have been established. We also provide the convergence rate of the variance estimation of the measurement error. Based on the results, the number of observations within each curve can be of any rate relative to the sample size, which is consistent with the earlier conclusions about the asymptotic properties of the mean and covariance estimators.

KEYWORDS:

• Functional principal components
• asymptotic properties
• local-linear smoothing
• weighing scheme

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work was supported by National Natural Science Foundation of China (project number: 11771146, 11831008, 81530086, 11771145), the National Social Science Foundation Key Program (17ZDA091), the 111 Project (B14019) and Program of Shanghai Subject Chief Scientist (14XD1401600). This work was also supported by the China Postdoctoral Science Foundation (2018M630393). We thank Professors Yanyuan Ma and Yukun Liu for very helpful discussions.

Notes on contributors

Xingyu Yan

Xingyu Yan is currently a Ph.D. candidate in the School of Statistics at East China Normal University. He is interested in functional data analysis.

Xiaolong Pu

Xiaolong Pu is currently Professor of statistics in the School of Statistics at East China Normal University. He is interested in applied statistics, particularly in statistical testing via sampling, statistical process control, sequential analysis and reliability. He received a Ph.D. in Statistics from East China Normal University.

Yingchun Zhou

Yingchun Zhou is currently Professor of statistics in the School of Statistics at East China Normal University. She is interested in functional data analysis and biostatistics. She received a Ph.D. in Statistics from Boston University and worked as a postdoc at National Institute of Statistical Sciences before moving to East China Normal University.

Xiaolei Xun

Xiaolei Xun is currently holding a visiting position in the School of Data Science at Fudan University. She is interested in functional data analysis, Bayesian methods, modeling of high-dimensional data and biostatistics. She received a Ph.D. in Statistics from Texas A&M University and worked in Novartis for five years as biometrician and statistical methodologist before moving to Fudan University.