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October 2011 Optimal estimation of the mean function based on discretely sampled functional data: Phase transition
T. Tony Cai, Ming Yuan
Ann. Statist. 39(5): 2330-2355 (October 2011). DOI: 10.1214/11-AOS898

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.

Citation

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T. Tony Cai. Ming Yuan. "Optimal estimation of the mean function based on discretely sampled functional data: Phase transition." Ann. Statist. 39 (5) 2330 - 2355, October 2011. https://doi.org/10.1214/11-AOS898

Information

Published: October 2011
First available in Project Euclid: 30 November 2011

zbMATH: 1231.62040
MathSciNet: MR2906870
Digital Object Identifier: 10.1214/11-AOS898

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.39 • No. 5 • October 2011
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