Solved: Suppose that the two-dimensional vectors (X1, Y1),

Chapter 7, Problem 24

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Suppose that the two-dimensional vectors (X1, Y1), (X2, Y2), . . . , (Xn, Yn) form a random sample from a bivariate normal distribution for which the means of X and Y , the variances of X and Y , and the correlation between X and Y are unknown. Show that the M.L.E.s of these five parameters are as follows: 1 = Xn and 2 = Y n, 52 1 = 1 n n i=1 (Xi Xn) 2 and 52 2 = 1 n n i=1 (Yi Y n) 2, = n i=1(Xi Xn)(Yi Y n) #n i=1(Xi Xn)2 $1/2 #n i=1(Yi Y n)2 $1/2 . Hint: First, rewrite the joint p.d.f. of each pair (Xi, Yi) as the product of the marginal p.d.f. of Xi and the conditional p.d.f. of Yi given Xi. Second, transform the parameters to 1, 2 1 and = 2 21 1 , = 2 1 , 2 2.1 = (1 2)2 2 . Third, maximize the likelihood function as a function of the new parameters. Finally, apply the invariance property of M.L.E.s to find the M.L.E.s of the original parameters. The above transformat