CONSISTENCY OF RESTRICTED MAXIMUM LIKELIHOOD
ESTIMATORS OF PRINCIPAL COMPONENTS
BY
DEBASHIS PAUL AND JIE PENG
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 [A geometric approach to maximum
likelihood estimation of covariance kernel from sparse irregular longitudinal
data (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.
Keywords: Functional
data analysis, principal component analysis, high-dimensional data, Stiefel manifold, intrinsic geometry, consistency.