Geodesic
Tests of independence of normal random variables with known and unknown variance ratio
Discussiones Mathematicae. Probability and Statistics, Tome 20 (2000) no. 2, pp. 233-247

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In the paper, a new approach to construction test for independenceof two-dimensional normally distributed random vectors is given under the assumption that the ratio of the variances is known. This test is uniformly better than the t-Student test. A comparison of the power of these two tests is given. A behaviour of this test forsome ε-contamination of the original model is also shown. In the general case when the variance ratio is unknown, an adaptive test is presented. The equivalence between this test and the classical t-test for independence of normal variables is shown. Moreover, the confidence interval for correlation coefficient is given. The results follow from the unified theory of testing hypotheses both for fixed effects and variance components presented in papers [6] and [7].
Keywords: mixed linear models, variance components, correlation, quadratic unbiased estimation, testing hypotheses, confidence intervals 
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Gąsiorek, Edward; Michalski, Andrzej; Zmyślony, Roman. Tests of independence of normal random variables with known and unknown variance ratio. Discussiones Mathematicae. Probability and Statistics, Tome 20 (2000) no. 2, pp. 233-247. http://geodesic.mathdoc.fr/item/DMPS_2000_20_2_a5/