ASYMPTOTIC STUDY OF THE MULTIVARIATE FUNCTIONAL-MODEL IN THE CASE OF A RANDOM NUMBER OF OBSERVATIONS FOR EACH MEAN

We consider the multivariate linear and affine functional models for w hich 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 mea n is a random variable. Let V-n(E) be the sampling covariance matrix E xplained by the partition (also called the between covariance matrix) and M a symmetric positive definite p x p matrix that defines a quadra tic metric on R(p). The least squares estimation of the parameters of the model in (R(p),M) amounts to the diagonalization of V-n(E) 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 inve rse of the errors covariance matrix Gamma(-1). When Gamma is unknown a nd estimated by the sampling Residual covariance matrix V-n(R) (also c alled the within covariance matrix), we are led to the diagonalization of V-n(E)(V-n(R))(-1) or V-n(E) V-n(-1) (with V-n = V-n(E) + V-n(R)). A study of the asymptotic properties of the estimators is then feasib le in the framework of Discriminant Factorial Analysis, in the case wh ere the population between covariance matrix V-E is assumed to be of r ank q.