Consistency of restricted maximum likelihood estimators of principal components

TL;DR

This paper establishes the consistency and convergence rates of REML estimators for principal components in both functional data and high-dimensional Gaussian vectors, highlighting their near-optimality and asymptotic equivalence.

Contribution

It extends the theoretical understanding of REML estimators by proving their consistency, convergence rates, and asymptotic properties in functional and high-dimensional settings.

Findings

REML estimators are consistent for eigenvalues and eigenfunctions under smoothness conditions.

Convergence rates of REML estimators are near-optimal when measurements per sample are bounded.

Asymptotic equivalence is shown between dense functional data inference and high-dimensional Gaussian vector analysis.

Abstract

In this paper we consider two closely related problems : estimation of eigenvalues and eigenfunctions of the covariance kernel of functional data based on (possibly) irregular measurements, and the problem of estimating the eigenvalues and eigenvectors of the covariance matrix for high-dimensional Gaussian vectors. In Peng and Paul (2007), a restricted maximum likelihood (REML) approach has been developed to deal with the first problem. In this paper, we establish consistency and derive rate of convergence of the REML estimator for the functional data case, under appropriate smoothness conditions. Moreover, we prove that when the number of measurements per sample curve is bounded, under squared-error loss, the rate of convergence of the REML estimators of eigenfunctions is near-optimal. In the case of Gaussian vectors, asymptotic consistency and an efficient score representation of the estimators are obtained under the assumption that the effective dimension grows at a rate slower than the sample size. These results are derived through an explicit utilization of the intrinsic geometry of the parameter space, which is non-Euclidean. Moreover, the results derived in this paper suggest an asymptotic equivalence between the inference on functional data with dense measurements and that of the high dimensional Gaussian vectors.