2.6. Let (Uį, Vį) be i.i.d. bivariate normal random vectors. The sample correla- tion coefficient is given by ô = suv/(s

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2 6 Let Ui Vi Be I I D Bivariate Normal Random Vectors The Sample Correla Tion Coefficient Is Given By O Suv S 1 (94.29 KiB) Viewed 49 times

2.6. Let (Uį, Vį) be i.i.d. bivariate normal random vectors. The sample correla- tion coefficient is given by ô = suv/(susv), where s² = n¯¹ [ï_1(U₂ − U)², s² n−¹ Σï_1(Vi – V)², suv n-1 n−¹ Σï_₁(Ui – U)(Vį – V), = = - U = n¯¹ Σ?_₁ U¿, and V = n¯¹ Σï-1 Vi. i=1 i=1 (a) Let xi = (Ui, Vi, U?, V², U¿V;)T, 1 ≤ i ≤ n, and x = n−1 Σ=1X. Show that p= g(ī), where x 5 - X1 X2 g((x1, T2, T3, T4, T5)T) (x3 – x²)¹/²(x₁ - x²)¹/² ° (b) Use (a) and the delta method to prove = √n(pp) → N(0, (1 - p²)²).