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

By Tommaso Cai, Ming Yuan. 2011

144 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/1bdf3d667400dd40f3bf07e16d01fe46793a30c9

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Citations in the corpus

Functional Data Analysis (2016, background): “The phenomenon of nonparametric versus parametric convergence rates as designs get more regular and denser have been characterized as a “phase transition” (Hall, M¨uller & Wang 2006; Cai & Yuan 2011).”
Peter Hall, functional data analysis and random objects (2016, background): “This was the first time a phase transition was observed to occur in the sampling schemes of functional data, with the discovery of additional phase transitions to follow later (Cai and Yuan, 2011).”
Nonparametric Covariance Function Estimation for Functional and Longitudinal Data (2010, unknown): “In particular, Cai and Yuan (2010) investigated the optimal rates for estimating µ0 under various settings.”
Asymptotic optimality and efficient computation of the leave-subject-out cross-validation (2012, result): “As a special case, recent results for the longitudinal function estimation by Cai and Yuan (2011) indicate that condition 3 would be satisfied in this particular setting if ξ(Σ,W)/n∗ = O(1) and n∗/n1/2r → 0 or ξ(Σ,W)/n∗ = o(1) and n∗/n1/2r → ∞ for some r > 1, where n∗ = ( 1n ∑n i=1 1 ni )−1 is the…”