Optimal Estimation of the Mean Function Based on Discretely Sampled Functional Data: Phase Transition
Tony Cai and Ming Yuan
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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 the common and independent designs. Minimax rates
of convergence are established and 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. Furthermore, our results show that for sparsely sampled functions, the independent design leads to superior convergence rate when compared to the common design, and therefore should be preferred in practice.
- Paper:
pdf file.
- Other related papers:
Cai, T. & Hall, P. (2006).
Prediction in functional linear regression.
The Annals of Statistics 34, 2159-2179.Cai, T. & Zhou, H. (2008).
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Technical report.Yuan, M. & Cai, T. (2010).
A reproducing kernel Hilbert space approach to functional linear regression.
The Annals of Statistics , to appear.