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The likelihood-ratio test for the hypothesis that the smallest of two canonical correlations is zero is Bartlett adjustable. This theoretical property says that the moments of the test criterion and of a scaled version of the asymptotic distribution have the same second-order expansion. Simulations show that scaling with the exact expectation improves the asymptotic distribution, but the expectation is approximated poorly by its second-order expansion. This can be explained by the asymptotic non-similarity of the test: a standard asymptotic distribution applies whenever the largest canonical correlation is nonzero whereas a nonstandard distribution applies in the case of complete independence. The latter distribution is described. Although the expressions for the two distributions are quite different, their quantiles are nearly proportional, explaining why the Bartlett adjustment works in practice despite a lack of similarity. Further, an accurate approximation is given for the expectation using an asymptotic technique which combines the two limit distributions continuously.
Biometrika © 1999 Biometrika Trust