The Annals of Statistics

Optimal estimation of the mean function based on discretely sampled functional data: Phase transition

T. Tony Cai and Ming Yuan

Full-text: Open access

Abstract

The problem of estimating the mean of random functions based on discretely sampled data arises naturally in functional data analysis. In this paper, we study optimal estimation of the mean function under both common and independent designs. Minimax rates of convergence are established and easily implementable rate-optimal estimators are introduced. The analysis reveals interesting and different phase transition phenomena in the two cases. Under the common design, the sampling frequency solely determines the optimal rate of convergence when it is relatively small and the sampling frequency has no effect on the optimal rate when it is large. On the other hand, under the independent design, the optimal rate of convergence is determined jointly by the sampling frequency and the number of curves when the sampling frequency is relatively small. When it is large, the sampling frequency has no effect on the optimal rate. Another interesting contrast between the two settings is that smoothing is necessary under the independent design, while, somewhat surprisingly, it is not essential under the common design.

Article information

Source
Ann. Statist., Volume 39, Number 5 (2011), 2330-2355.

Dates
First available in Project Euclid: 30 November 2011

Permanent link to this document
https://projecteuclid.org/euclid.aos/1322663460

Digital Object Identifier
doi:10.1214/11-AOS898

Mathematical Reviews number (MathSciNet)
MR2906870

Zentralblatt MATH identifier
1231.62040

Keywords
Functional data mean function minimax rate of convergence phase transition reproducing kernel Hilbert space smoothing splines Sobolev space

Citation

Cai, T. Tony; Yuan, Ming. Optimal estimation of the mean function based on discretely sampled functional data: Phase transition. Ann. Statist. 39 (2011), no. 5, 2330--2355. doi:10.1214/11-AOS898. https://projecteuclid.org/euclid.aos/1322663460


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