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Asymptotic study of the multivariate functional model. Application to the metric choice in principal component analysis. (English) Zbl 0809.62052
Summary: The least squares estimation of the parameters of the functional models in \((\mathbb{R}^ p, M)\), where \(M\) is a symmetric positive definite \(p \times p\) matrix that defines a quadratic metric on \(\mathbb{R}^ p\), amounts to a Principal Component Analysis (PCA) of order \(q\) in \((\mathbb{R}^ p, M)\). We assume that the errors are independent and have identical moments up to order 6. We study the almost sure convergence of the estimators and prove that they are consistent if and only if \(M = k \Gamma^{-1}\) (\(k > 0\)) where \(\Gamma\) is the known covariance matrix of the errors. This result is a property of a Gauss-Markov type for PCA and gives insight into the choice of metric in PCA. We study the asymptotic distributions of these estimators. When \(M = \Gamma^{-1}\) and the errors are elliptical, in particular Gaussian, we give explicitly the covariance operators of the Gaussian limiting distributions and show applications to statistical inference.

MSC:
62H25 Factor analysis and principal components; correspondence analysis
62E20 Asymptotic distribution theory in statistics
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