CLT and Confidence Limits | ISI MStat 2016 PSB Problem 8

This is a problem from ISI MStat Examination 2016. This primarily tests the student's knowledge in finding confidence intervals and using the Central Limit Theorem as a useful approximation tool.

The Problem:

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

(a) Show that there exists a function $c(\rho)$ such that :

$\lim_{n \rightarrow \infty} P(\sqrt{n}(\bar{X}-\bar{Y}) \le c(\rho) )=\Phi(1)$ , where $\Phi$ is the cdf of $N(0,1)$

(b)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
$\lim_{n \rightarrow \infty} P(L_n<\rho<1)=\alpha$ .

Prerequisites:

(a)The Central Limit Theorem and Convergence of Sequence of RVs

(b)Idea of pivots and how to obtain confidence intervals.

Solution:

(a) See that $E(\bar{X}-\bar{Y})=0$ .

See that $\text{Var}(\bar{X}-\bar{Y})=\frac{\sum_{i=1}^{n} \text{Var}(X_i-Y_i)}{n^2} =\frac{2(1-\rho)}{n}$

Take $c(\rho)=2(1-\rho)$ .

By the Central Limit Theorem,

see that $P(\frac{\sqrt{n}(\bar{X}-\bar{Y})}{\sqrt{2(1-\rho)}} \le 1) \rightarrow \Phi(1)$ as $n \rightarrow \infty$

(b) Let $Z_i = X_i - Y_i$

$Z_i$ 's are iid with $E(Z_i)=0$ and $V(Z_i)=2(1- \rho)$

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

Use this as the pivot to obtain an asymptotic confidence interval for $\rho$ .

See that $P(\frac{\sqrt{n} \bar{Z}}{\sqrt{2-2\rho}}) \ge \tau_{\alpha})=\alpha$ , where $\tau_\alpha$ : upper $\alpha$ point of $N(0,1)$ .

Equivalently , you can write, $P( \rho \ge 1- \frac{n \bar{Z}^2 }{2 \tau_{\alpha}^2} ) =\alpha$ , as $n \rightarrow \infty$ .

Thus , $L_n=1- \frac{n \bar{Z}^2 }{2 \tau_{\alpha}^2}$ .

Food For Thought:

Suppose $X_1,..,X_n$ are equi-correlated with correlation coefficient $\rho$ .

Given, $E(X_i)= \mu$ , $V(X_i) = \sigma_i ^2$ , for $i=1,2,..,n$ .

Can you see that $\rho \ge -\frac{1}{\frac{(\sum \sigma_i)^2}{\sum \sigma_i ^2} -1 }$ ?

It's pretty easy right? Yeah, I know 😛 . I am definitely looking forward to post inequalities more often .

This may look trivial but it is a very important result which can be used in problems where you need a non-trivial lower bound for $\rho$ .

Well,well suppose now $Y_i=X_i - \bar{X}$ .

Can you show that a necessary and sufficient condition for $X_i$ , $\{ i=1,2,..,n \}$ to have equal variance is that $Y_i$ should be uncorrelated with $\bar{X}$ ?

Till then Bye!

Stay Safe.