 Solution

With $\Sigma =I_d$, constrained likelihood maximization minimizes $\Vert \theta -\overline{X}{\Vert }^2$. On the unit sphere this is $1+\Vert \overline{X}{\Vert }^2-2{\theta }^T\overline{X}$. By the Cauchy-Schwarz inequality its minimizer isIf $\overline{X}=0$, every unit vector is a maximum-likelihood estimator; a specified measurable tie-breaking choice makes an estimator defined everywhere. This event has probability zero in the nonsingular normal model, but the optimization still has that exceptional case.