Abstract
The common principal components (CPC) model for several groups of multivariate observations assumes equal principal axes but possibly different variances along these axes among the groups. Under a CPCs model, generalized projection-pursuit estimators are defined by using score functions on the dispersion measure considered. Their partial influence functions are obtained and asymptotic variances are derived from them. When the score function is taken equal to the logarithm, it is shown that, under a proportionality model, the eigenvector estimators are optimal in the sense of minimizing the asymptotic variance of the eigenvectors, for a given scale measure.
Highlights
Several authors, as [10], have studied models for common structure dispersion
The more restrictive proportionality model assumes that the scatter matrices are equal up to a proportionality constant, i.e., i = i 1 for 1 i k and 1 = 1
Partial influence functions were introduced by [14] in order to ensure that the usual properties of the influence function for the one-population case are satisfied when dealing with several populations
Summary
As [10], have studied models for common structure dispersion As it is well known, those models have been introduced to overcome the problem of an excessive number of parameters, when dealing with several populations, in multivariate analysis. One such basic common structure assumes that the k covariance matrices have possibly different eigenvalues but identical eigenvectors, i.e., i = i , 1 i k,. In [10] a unified study of the maximum likelihood estimators under a CPC model and, in particular, under a proportionality model is given.
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