Suppose that the observations \(Y_{1}, \ldots, Y_{n}\) in a two-arm randomized design are independent bivariate Gaussian with mean vector \(\left(\mu_{1}, \mu_{2}ight)\) for units in the control arm, and

\[
E\left(Y_{i} \mid \mathbf{t}ight)=\left(\begin{array}{l}
\mu_{1} \cos \tau-\mu_{2} \sin \tau \\
\mu_{1} \sin \tau+\mu_{2} \cos \tau
\end{array}ight)
\]

for units in the active treatment arm. The covariance in both cases is \(\sigma^{2} I_{2}\); the parameter \(\mu \in \mathbb{R}^{2}\) is unrestricted, while \(\sigma>0\) and the treatment effect lies in \(0 \leq \tau<2 \pi\). By expressing the sample averages for each treatment arm as complex numbers, show that \(\hat{\tau}=\arg \left(\bar{Y}_{0}ight)-\arg \left(\bar{Y}_{1}ight)\) is the maximum-likelihood estimate of the treatment effect. Find the maximum-likelihood estimate of \(\sigma^{2}\).