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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.

62H25 Factor analysis and principal components; correspondence analysis
62E20 Asymptotic distribution theory in statistics
Full Text: DOI
[1] DOI: 10.1214/aos/1176346390 · Zbl 0542.62039 · doi:10.1214/aos/1176346390
[2] Arconte A., C.R.Ac.Sc.Paris pp 319– (1980)
[3] Besse P., Compstat 1986 (1986)
[4] Besse P., C.R.Ac.Sc.Paris pp 319– (1987)
[5] DOI: 10.1080/02331888808802103 · Zbl 0643.62046 · doi:10.1080/02331888808802103
[6] Bunke H., Statistical Inference in Linear Models 1 (1986) · Zbl 0579.62051
[7] Caussinus H., Multidimensional Data Analysis pp 149– (1985)
[8] Caussinus H., Data Analysis and Informatics IV pp 151– (1985)
[9] DOI: 10.1016/0047-259X(82)90088-4 · Zbl 0539.62064 · doi:10.1016/0047-259X(82)90088-4
[10] DOI: 10.1080/02331889108802329 · Zbl 0737.60055 · doi:10.1080/02331889108802329
[11] Ferre, L. 1988. ”Problèmes de représentation optimale par l’A.C.P. Thèse”. Toulouse: Université Paul Sabatier.
[12] DOI: 10.1080/02331888708802037 · Zbl 0647.62055 · doi:10.1080/02331888708802037
[13] DOI: 10.1080/03610929408801801 · Zbl 0616.62080 · doi:10.1080/03610929408801801
[14] Gedler G., Etude asymptotique (1986)
[15] Malinvaud E., Méthodes mathématiques de l’économétric (1981)
[16] DOI: 10.1002/9780470316559 · doi:10.1002/9780470316559
[17] Renyi A., Calcul des Probabilités (1966)
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