Let (Ui, Vi) be i.i.d. bivariate normal random vectors.
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Let (Ui, Vi) be i.i.d. bivariate normal random vectors. The sample correlation coefficient is given by ? = sUV (sUsV ),
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Let (Ui, Vi) be i.i.d. bivariate normal random vectors. The sample correlation coefficient is given by ? = sUV (sUsV ), where s2 U = n?1 n i=1(Ui ? U)2, s2 V = n?1 n i=1(Vi ? V )2, sUV = n?1 n i=1(Ui ? U)(Vi ? V ), U = n?1 n i=1 Ui, and V = n?1 n i=1 Vi. (a) Let xi = (Ui, Vi, Ui2, Vi2, UiVi)T, 1 ? i ? n, and x = n?1 n i=1 xi. Show that ? = g(x), where g((x1, x2, x3, x4, x5)T) = x5 ? x1x2 (x3 ? x2 1)1/2(x4 ? x2 2)1/2. (b) Use (a) and the delta method to prove ?n(? ? ?) ? N(0, (1 ? ?2)2).
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