Nonparametric Test for Eigenvalues of Covariance Matrix in Multipopulation

Abstract

We propose a nonparametric procedure to test the hypothesis that the j-th largest eigenvalues of a covariance matrix are equal in multipopulation. We apply the Mood test by using the principal component scores and deal the equality of eigenvalues with the equality of variance. We investigate the significance level and the power of test by simulation and show that this nonparametric test is useful for symmetric populations.

The modified Mood test for the scale alternative and its numerical comparisons

“…However, it is very difficult to obtain the precise distribution of eigenvalues of a covariance matrix for a non-normal population. Murakami, Hino, and Tsukada (2007) proposed a procedure to test the equality of eigenvalues that used principal component scores by applying the Mood test. Additionally, unreplicated twolevel fractional factorial designs are often used in industry to study the effects of factors on the mean of a response.…”

A Statistic for Testing the Equality of Eigenvalue of Covariance Matrix on Multipopulation

“…In addition, Takeda (2001) proposed a test statistic and derived the exact distribution of the statistic under normality. Recently, Murakami, Hino and Tsukada (2007) extended the testing procedure, which uses two nonparametric tests, from a bipopulation to a multipopulation. However, we need large sample sizes to obtain high power when we use a nonparametric test.…”

“…Therefore we set the critical value of both test as 5.991 for k = 3. Murakami et al (2007) show how the statistic AB3k and the statistic M k are used to test the hypothesis. We simulate 100,000 times for the statistics AB2k and Mk, and investigate the type I error and the power of the test using the statistics T~ , ABA k and M3k.…”

“…We set the populations as normal populations N3 (0, E') and contaminated normal populations 0.95 x N3 (0, E~2)) + 0.05 x N3 (0, and the sample sizes are N1 = N2 = N3 = 50, N1 = N2 = N3 = 100 and N1 = 200, N2 = 150, N3 = 100. We treat the same cases as Murakami et al (2007) and add the case of N1 = 150, N2 = 100, N3 = 50 to compare the type I error and the power for the case of N1 = N2 = N3 = 100. Since the distribution of the eigenvalues may not depend on the eigenvector, it is sufficient to examine only the case that the population covariance matrices are diagonal without loss of generality.…”