Inference under functional proportional and common principal component models

In many situations, when dealing with several populations with different covariance operators, equality of the operators is assumed. Usually, if this assumption does not hold, one estimates the covariance operator of each group separately, which leads to a large number of parameters. As in the multivariate setting, this is not satisfactory since the covariance operators may exhibit some common structure. In this paper, we discuss the extension to the functional setting of common principal component model that has been widely studied when dealing with multivariate observations. Moreover, we also consider a proportional model in which the covariance operators are assumed to be equal up to a multiplicative constant. For both models, we present estimators of the unknown parameters and we obtain their asymptotic distribution. A test for equality against proportionality is also considered. Some key words: Common principal components, Eigenfunctions, Functional data analysis, Hilbert-Schmidt operators, Kernel methods, Proportional model. Corresponding Author Graciela Boente Moldes 1855, 3 A Buenos Aires, C1428CRA Argentina email: [email protected] fax 54-11-45763375 Running Head: Functional common principal component model.