# One-sided confidence interval for correlation coefficient

Consider this question,

Let $$(X_1,Y_1),(X_2,Y_2),...,(X_n,Y_n)$$ be independent and identically distributed pairs of random variables with $$E(X_1) = E(Y_1), Var(X_1)= Var(Y_1) = 1$$, and $$Cov(X_1,Y_1) = \rho \in (-1,1).$$

Given $$\alpha \in (0,1)$$, obtain a statistic $$L_n$$ which is a function of $$(X_1,Y_1),(X_2,Y_2),...,(X_n,Y_n)$$ such that \begin{align} \displaystyle \lim_{n\to\infty}P(L_n < \rho < 1) = \alpha \end{align}

My concern is that since it is given that $$\rho \in (-1,1)$$, can I drop the $$1$$ in the equality to be proven. I mean, will the statistic $$L_n$$ calculated such that it satisfies \begin{align} \displaystyle \lim_{n\to\infty}P(L_n < \rho) = \alpha \end{align} would also satisfy the given equation. And if not, how should I go about this problem?

• – StubbornAtom May 1 at 10:10
• I had calculated it the same way once I dropped off the $1$ from the equation. However, I want to ask that can I do that? – Sanket Agrawal May 1 at 10:42
• The question asks for a lower confidence bound for the correlation; it is of course bounded above by $1$, whether you mention it or not. – StubbornAtom May 1 at 10:46
• Okay, got your point. Thanks – Sanket Agrawal May 1 at 13:49