(i) If X1,..., Xn; Y1,..., Yn are independent normal variables with common variance 2 and means E(Xi)...

(i) If X1,..., Xn; Y1,..., Yn are independent normal variables with common variance σ2 and means E(Xi) = ξi , E(Yi) = ξi + , the UMP unbiased test of = 0 against > 0 is given by (5.58).

(ii) Determine the most accurate unbiased confidence intervals for .

[(i): The structure of the problem becomes clear if one makes the orthogonal transformation X

i = (Yi − Xi)/√2, Y 

i = (Xi + Yi)/√2.]