(i) Let (X1, Y1),..., (Xn, Yn) be a sample from the bivariate normal distribution (5.69), and let S2 1 = (Xi − X¯)

2, S2 2 = (Yi − Y¯ )

2, S12 = (Xi − X¯)(Yi − Y¯ ). There exists a UMP unbiased test for testing the hypothesis τ /σ = ∆. Its acceptance region is

|∆2S2 1 − S2 2 |

(∆2S2 1 + S2 2 )2 − 4∆2S2 12

≤ C, and the probability density of the test statistic is given by (5.83) when the hypothesis is true.

(ii) Under the assumption τ = σ, there exists a UMP unbiased test for testing η = ξ, with acceptance region |Y¯ −X¯|/
S2 1 + S2 2 − 2S12 ≤ C. On multiplication by a suitable constant the test statistic has Student’s t-distribution with n − 1 degrees of freedom when η = ξ.
[Due to Morgan (1939) and Hsu (1940). (i): The transformation U = ∆X + Y , V = X − (1/∆)Y reduces the problem to that of testing that the correlation coefficient in a bivariate normal distribution is zero.
(ii): Transform to new variables Vi = Yi − Xi, Ui = Yi + Xi.]