Asymptotic study of the multivariate functional model in the case of a random number of observations for each mean

Abstract

We consider the multivariate linear and affine functional models for which several observations for each mean are available (replications of observations). In the case of a simple random sampling, which is the assumption made in this study, the number of observations for each mean is a random variable. Let be the sampling covariance matrix Explained by the partition (also called the between covariance matrix) and M a symmetric positive definite p × p matrix that defines a quadratic metric on . The least squares estimation of the parameters of the model in ( , M)amounts to the diagonalization of M The estimators are consistent for any M, but we show that they satisfy an asymptotic efficiency property if and only if we choose for M the inverse of the errors covariance matrix Γ-1. When Γ is unknown and estimated by the sampling Residual covariance matrix (also called the within covariance matrix), we are led to the diagonalization of or (with . A study of the asymptotic properties of the estimators is then feasible in the framework of Discriminant Factorial Analysis, in the case where the population between covariance matrix V E is assumed to be of rank q.