Exercise 26 (#2.58). Suppose that (X1, Yl),..., (Xn, Yn) are independent and identically distributed random 2-vectors having the normal distribution with EX1 = EY1 = 0, Var(X1) = Var(Yi) = 1, and Cov(X1, Y1) = 0 e (-1, 1). (i) Find a minimal sufficient statistic for 0. (ii) Show whether the minimal sufficient statistic in (i) is complete or not. 68 Chapter 2. Fundamentals of Statistics (iii) Prove that Ti = Et_, X? and T2 = Et, Y? are both ancillary but (T1, T2) is not ancillary. Solution. (i) The joint Lebesgue density of (X1, Yl),..., (Xn, Yn) is n n 20 n 27VI - 02 exp 1 - 0 [(x7 + y?) + Liyi i=1 i=1 Let 7 = 1 20 1 - 02'1 - 02 The parameter space {7 : -1