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June 2009 Consistency of restricted maximum likelihood estimators of principal components
Debashis Paul, Jie Peng
Ann. Statist. 37(3): 1229-1271 (June 2009). DOI: 10.1214/08-AOS608


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.


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Debashis Paul. Jie Peng. "Consistency of restricted maximum likelihood estimators of principal components." Ann. Statist. 37 (3) 1229 - 1271, June 2009.


Published: June 2009
First available in Project Euclid: 10 April 2009

zbMATH: 1161.62032
MathSciNet: MR2509073
Digital Object Identifier: 10.1214/08-AOS608

Primary: 60K35 , 62G20
Secondary: 62H25

Keywords: consistency , Functional data analysis , High-dimensional data , Intrinsic geometry , Principal Component Analysis , Stiefel manifold

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 3 • June 2009
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