ABSTRACT

The problem of testing the hypothesis H0 : ?1=···=?p=s 2 has been treated in Chapter

8 under the title sphericity test for Np(µ, S) and Ep(µ, S). If H0 is accepted we conclude that the principal components all have the same variance and hence contribute equally to the total variation. Thus no reduction in dimension can be achieved by transforming the variable to its principal components. On the other hand if H0 is rejected it is natural to test the hypothesis that ?2=···=?p and so on. Theorem 10.6.1 gives the likelihood ratio test of H0 : ?k+1=···=?p=? where ? is unknown for Np(µ, S). For elliptically symmetric distributions similar results can be obtained using Theorem 5.3.6.