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Let $(X_1,Y_1),(X_2,Y_2),..,(X_n,Y_n)$ be iid pairs of random variables with $E(X_1)=E(Y_1)$, $\text{Var}(X_1)=\text{Var}(Y_1)=1$,and $\text{cov}(X_1,Y_1)=\rho \in(-1,1)$. Given $\alpha>0$ , obtain a statistic $L_n$ which is a function of $(X_1,Y_1),(X_2,Y_2),..,(X_n,Y_n)$ such that $\lim_{n \rightarrow \infty} P(L_n<\rho<1)=\alpha$.

Since the sample distribution of the correlation coefficient is quite hard to work with I am really not being able to crack this one. Can anyone help?

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Let $Z_i=X_i-Y_i$, so that $Z_1,Z_2,\ldots,Z_n$ are i.i.d variables with zero mean and variance $2-2\rho$.

So by classical CLT,

$$\frac{\sqrt n\overline Z}{\sqrt{2-2\rho}}\stackrel{L}\longrightarrow N(0,1)\,,$$

where $\overline Z=\frac{1}{n} \sum\limits_{i=1}^n Z_i=\overline X-\overline Y$.

Using this pivot, you can readily obtain an asymptotic confidence interval for $\rho$.

In particular, you can start from $$\frac{\sqrt n\overline Z}{\sqrt{2-2\rho}}\ge z_{\alpha}$$

to get a lower bound for $\rho$ with limiting probability $\alpha$.

Here $z_{\alpha}$ is of course the $(1-\alpha)$th fractile of $N(0,1)$.

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