Abstract
SUMMARY Exact and approximate likelihood ratio tests are derived for certain structures in multivariate normal correlation matrices. Reasonable asymptotic distributions for the approximate tests are proposed and examined by empirical sampling. The powers of these tests are compared with those of previously proposed tests. Tests for certain structures in correlation matrices have been proposed by several authors. Hotelling (1940) proposed a conditional t test for the equality of two correlations in a trivariate normal distribution; Bartlett (1950, 1951), Anderson (1963) and Lawley (1963) considered tests for equality of all correlations in a multivariate normal distribution; and Bartlett & Rajalakshman (1953) and Kullback (1959) proposed a test for a completely specified correlation matrix. The object of the present investigation is to examine likelihood ratio tests, or simple approximations to them, for the above hypotheses. Asymptotic distributions are not available for the approximate tests, but reasonable approximations are proposed and examined by empirical sampling. The powers of these tests are compared with those of previously proposed tests.
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