Let p(x;$\theta$) be the density of bivariate normal $\textbf{X}$ whose components have unknown mean $\theta=(\theta_1,\theta_2)$, known variances $\sigma_1^2,\sigma_2^2$ and known correlation coefficient $\rho\in(-1,1)$. For testing $H_0:\theta_1=0,\theta_2=0$ vs $H_1:\theta\in\mathbb{R}^2-\{0,0\}$, show that $\lambda$~ $\chi_1^2$ where $-2log\lambda_n\rightarrow\lambda$ in distribution as $n\rightarrow\infty$ and $\lambda_n$ is the log likelihood ratio test statistic.

I firstly found the MLE of $\theta_1$ and $\theta_2$ as $\bar{X}$ and $\bar{Y}$. Then simplified $-2log(\lambda_n)$, and found

\begin{equation} -2log(\lambda_n)=\frac{n}{1-\rho^2}\left[ \left(\frac{\bar{x}}{\sigma_1}\right)^2-\frac{2\rho}{\sigma_1\sigma_2}\bar{x}\bar{y}+\left(\frac{\bar{y}}{\sigma_2}\right)^2 \right] \end{equation}

I got stuck at this point, and can't see how to proceed from here!


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