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].
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1014,
author = {Edward G\k asiorek and Andrzej Michalski and Roman Zmy\'slony},
title = {Tests of independence of normal random variables with known and unknown variance ratio},
journal = {Discussiones Mathematicae Probability and Statistics},
volume = {20},
year = {2000},
pages = {233-247},
zbl = {1123.62309},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1014}
}
Edward Gąsiorek; Andrzej Michalski; Roman Zmyślony. Tests of independence of normal random variables with known and unknown variance ratio. Discussiones Mathematicae Probability and Statistics, Tome 20 (2000) pp. 233-247. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1014/
[000] [1] S. Geisser, Estimation in the uniform covariance case, JASA 59 (1964), 477-483. | Zbl 0192.25902
[001] [2] S. Gnot and A. Michalski, Tests based on admissible estimators in two variance components models, Statistics 25 (1994), 213-223. | Zbl 0816.62019
[002] [3] N.L. Johnson and S. Kotz, Distribution in Statistics: continuous univariate distributions - 2, Houghton Mifflin, New York 1970.
[003] [4] J.M. Kinderman and J.F. Monahan, Computer generation of random variables using the ratio of uniform deviates, ACH Trans. Math. Soft. 3 (1977), 257-260. | Zbl 0387.65006
[004] [5] E.L. Lehmann, Testing Statistical Hypotheses, Wiley, New York 1986.
[005] [6] A. Michalski and R. Zmyślony, Testing hypotheses for variance components in mixed linear models, Statistics 27 (1996), 297-310. | Zbl 0842.62059
[006] [7] A. Michalski and R. Zmyślony, Testing hypotheses for linear functions of parameters in mixed linear models, Tatra Mountains Math. Publ. 17 (1999), 103-110. | Zbl 0987.62012
[007] [8] J. Seely, Quadratic subspaces and completeness, Ann. Math. Statist. 42 (1971), 710-721. | Zbl 0249.62067
[008] [9] R. Zmyślony, On estimation of parameters in linear models, Zastosowania Matematyki 15 (1976), 271-276. | Zbl 0401.62049
[009] [10] R. Zmyślony, Completeness for a family of normal distributions, Banach Center Publications 6 (1980), 355-357. | Zbl 0464.62003