Test of hypothesis on the means of a bivariate correlated normal

The purpose of this paper is to improve the procedure proposed by Duncan to test the hypotheses that the means of a Bivariate Correlated Normal (B.C.N.) are both equal to fixed values against all the possible alternatives. The Duncan test is based on the Bonferroni inequality that is the situation in which the components of the Bivariate Normal are independent. The procedure is simple to be treated but the resulting test is conservative. The alternative way of verifying the hypotheses in question proposed here takes into account the correlation coefficient between the two components. The test is based on the exact distributions of the absolute maximum, proposed by Zenga in 1979, and the absolute minimum of the components of a Standardized Bivariate Correlated Normal (S.B.C.N.) r.v. proposed in 2004 by Pollastri-Tornaghi. The two distributions, and especially that of the absolute minimum, are not widely known and so the critical values are reported in order to allow to the researcher to use the test here proposed. The new stepwise test is attractive for its simplicity and allows the acceptance of the null hypotheses when it is true with a fixed probability error rate, while Duncan's procedure is conservative. Moreover, the procedure here considered is more powerful than Duncan's.