Discussiones Mathematicae Probability and Statistics 20(2) (2000) 233-247

TESTS OF INDEPENDENCE OF NORMAL RANDOM VARIABLES WITH KNOWN AND UNKNOWN VARIANCE RATIO

Edward Gąsiorek, Andrzej Michalski

Department of Mathematics, Institute of Mathematics
Agriculture University of Wrocław
Grunwaldzka 53, 50-357 Wrocław, Poland

e-mail:max@ozi.ar.wroc.pl

and

Roman Zmyślony

Institute of Mathematics, Technical University
Podgórna 50, 65-246 Zielona Góra, Poland
e-mail:r.zmyslony@im.uz.zgora.pl

Abstract

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 and phrases: mixed linear models; variance components;correlation; quadratic unbiased estimation; testing hypotheses; confidence intervals.

1999 Mathematics Subject Classiffication: 62F03, 62J10.

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Received 5 September 2000