Robust tests for the common principal components model

In multivariate analysis we often deal with situations involving several populations, such as discriminant analysis, where the assumption of equality of scatter matrices is usually assumed. Yet sometimes, this assumption is not adequate but problems related to an excessive number of parameters will arise if we estimate the scatter matrices separately for each population. In many practical situations this problem can be avoided if the scatter matrices of the different populations exhibit some common structure. Several authors, as for instance Flury (1988), have studied models for common structure dispersion. One such basic common structure assumes that the k scatter matrices have different eigenvalues but identical eigenvectors, i.e., the matrices are pairwise commutable and may be simultaneously diagonalizable. That is, there is a (p × p)-orthogonal matrix β : Σi = βΛiβ, i = 1, . . . , k where Σi is the (p × p)-scatter matrix of the i−th population and Λi = diag(λi1, ..., λip). This model, proposed by Flury (1984), became known as the Common Principal Components (cpc) model. The more restrictive proportionality model assumes that the scatter matrices are equal up to a proportionality constant, i.e., Σi = ρiΣ1 , for 1 ≤ i ≤ k and ρ1 = 1. In Flury (1988) a unified study of the maximum likelihood estimators under a cpc model and under a proportionality model is given and likelihood ratio tests for a hierarchy of models is studied. However, as it is well known, the likelihood ratio test are in most situations affected by anomalous observations. In this work we propose robust procedures for testing the relationship between scatter matrices. An robust statistic for testing proportionality against a common principal components model is considered. Also, the null hypothesis of a cpc model versus no restrictions on the scatter matrices is studied.