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

TL;DR

This paper investigates the optimal estimation of the mean function from discretely sampled functional data, revealing phase transition phenomena influenced by sampling design and frequency, and proposing rate-optimal estimators.

Contribution

It introduces easily implementable estimators and characterizes phase transition phenomena in the convergence rates under different sampling designs.

Findings

Sampling frequency determines convergence rate under common design when small.

Under independent design, convergence rate depends on both sampling frequency and number of curves.

Smoothing is necessary under independent design but not under common design.

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.