Asympotic study of the multivariate functional model. application to the metric choice in principal component analysis

摘要

The least squares estimation of the parameters of the functional models in (,M) whereMis a symmetric positive definitep#xD7;pmatrix that defines a quadratic metric on (, amounts to a Principal Component Analysis (PCA) of orderqin (,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 ifM=k#x393;-1(k 0) where #x393; is the known covariance matrix of the errors. This result is a property of a Gauss-Markov type for PCA and give insight into the choice of metric in PCA. We study the asymptotic distributions of these estimators. WhenM= #x393;-1and the errors are elliptical, in particular Gaussian, we give explicitly the covariance operators of the Gaussian limiting distributions and show applications to statistical inference.