For $i=1, \ldots, n$, the pairs $\left(X_{i}, Y_{i}\right)$ have independent bivariate normal distributions, with $\mathbb{E}\left(X_{i}\right)=\mu_{X}, \mathbb{E}\left(Y_{i}\right)=\mu_{Y}, \operatorname{var}\left(X_{i}\right)=\operatorname{var}\left(Y_{i}\right)=\phi$, and $\operatorname{corr}\left(X_{i}, Y_{i}\right)=\rho$. The means $\mu_{X}, \mu_{Y}$ are known; the parameters $\phi>0$ and $\rho \in(-1,1)$ are unknown.

Show that the joint distribution of all the variables belongs to an exponential family, and identify the natural sufficient statistic, natural parameter, and mean-value parameter. Hence or otherwise, find the maximum likelihood estimator $\hat{\rho}$ of $\rho$.

Let $U_{i}:=X_{i}+Y_{i}, V_{i}:=X_{i}-Y_{i}$. What is the joint distribution of $\left(U_{i}, V_{i}\right) ?$

Show that the distribution of

$$\frac{(1+\hat{\rho}) /(1-\hat{\rho})}{(1+\rho) /(1-\rho)}$$

is $F_{n}^{n}$. Hence describe a $(1-\alpha)$-level confidence interval for $\rho$. Briefly explain what would change if $\mu_{X}$ and $\mu_{Y}$ were also unknown.

[Recall that the distribution $F_{\nu_{2}}^{\nu_{1}}$ is that of $\left(W_{1} / \nu_{1}\right) /\left(W_{2} / \nu_{2}\right)$, where, independently for $j=1$ and $j=2, W_{j}$ has the chi-squared distribution with $\nu_{j}$ degrees of freedom.]