Bent Nielsen
  (1999)
  'The likelihood ratio test for rank in bivariate canonical
   correlation analysis'
  Biometrika 86, Issue 2

  ABSTRACT:
  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 non-zero whereas a
  non-standard distribution applies in 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.