Statistics Papers

Document Type

Journal Article

Date of this Version

2011

Publication Source

The Annals of Statistics

Volume

39

Issue

5

Start Page

2330

Last Page

2355

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.

Keywords

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

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Date Posted: 27 November 2017

This document has been peer reviewed.