CONSISTENCY OF RESTRICTED MAXIMUM LIKELIHOOD ESTIMATORS OF PRINCIPAL COMPONENTS

By Debashis Paul, Jie Peng. 2008

49 citations recorded by the source corpus.

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 [23], a restricted maximum likelihood (REML) approach has been developed to deal with the flrst 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 e‐cient score representation of the estimators are obtained under the assumption that the efiective 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.

Subfields: Economics and Econometrics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/734a4bd20aed30a868617d7091390f7abf61bb80

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Economics and Econometrics

In Statistics and Probability

Citations in the corpus

Functional Data Analysis (2016, methodology): “For the much more diﬃcult but commonly encountered situation of sparse functional data, the FPCA approach was investigated in Shi, Weiss & Taylor (1996); Staniswalis & Lee (1998); James, Hastie & Sugar (2000); Rice & Wu (2001); Yao, M¨uller & Wang (2005a); Yao & Lee (2006); and Paul & Peng (2009).”
A Smooth Simultaneous Confidence Corridor for the Mean of Sparse Functional Data (2014, methodology): “Except normality of the ξik’s, Assumption (A3) has been used in Yao, Müller, and Wang (2005a) and Paul and Peng (2009), while Assumption (A4) is in Ma, Yang, and Carroll (2012).”
A Geometric Approach to Maximum Likelihood Estimation of the Functional Principal Components From Sparse Longitudinal Data (2007, background): “A first-order expansion of the difference between the averaged loss under the true and estimated parameters shows that the Kullback–Leibler loss correctly scales the difference, whereas the prediction error loss does not (Paul and Peng 2008).”
Nonparametric Covariance Function Estimation for Functional and Longitudinal Data (2010, unknown): “It is also interesting to compare our results with those obtained by Paul and Peng (2009) who consider a particular setting where C0 can be approximated by a function with a fixed number of non-vanishing eigenvalues and eigenfunctions from W42 .”
COVARIATE ADJUSTED FUNCTIONAL PRINCIPAL COMPONENTS ANALYSIS FOR LONGITUDINAL DATA (2010, background): “However, these difficulties can be overcome in most situations, so it is still possible to conduct FPCA [Shi, Weiss and Taylor (1996), James, Hastie and Suger (2000), Rice and Wu (2001), Yao, Müller and Wang (2005), Paul and Peng (2009) and Peng and Paul (2009)].”