Suppose that $X_1, X_2, \ldots, X_n \sim N_p(\mu_x, \Sigma_x)$. If we assume that $\mu_x = k_1\mu_0$ where $\mu_0$ is known and $\Sigma_x$ is known. Derive the maximum likelihood estimator of $k_1$. Is it unbiased?
Can we simply say here that since normally $\hat{\mu} = \bar{x},$ that now $k_1 = \frac{\hat{\mu}}{\mu_0}= \frac{\bar{x}}{\mu_0}?$.
Yes. More generally, invertible transformations of the parameters behave the obvious way under MLE. This is because the likelihood for a specific dataset is a function of the parameters only, so the point in parameter space that maximises that function is unchanged by coordinate transformations on that space.