Tests of independence of normal random variables with known and unknown variance ratio. (Q2718382)

In this paper, a new approach to the construction of tests for independence of 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 for some e-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 of \textit{A. Michalski} and \textit{R. Zmyślony} [Statistics 27, No. 3--4, 297--310 (1996; Zbl 0842.62059), Tatra Mt. Math. Publ. 17, 103--110 (1999; Zbl 0987.62012)].