Optimal Estimation of the Mean Function Based on Discretely Sampled Functional Data: Phase Transition
Tony Cai and Ming Yuan
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.
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