Statistical estimation

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Let $x^{(1)}, \ldots, x^{(N)}$ be independent samples from an $\mathcal N(\mu,\Sigma)$ distribution, where it is known that $\Sigma^\mathrm{min} \preceq \Sigma \preceq \Sigma^\mathrm{max}$, where $\Sigma^\mathrm{min},~ \Sigma^\mathrm{max}$ are given positive definite matrices.

The associated log-likelihood function $\ell(\mu,\Sigma)$ (including the constraint on $\Sigma$) is concave.

True.

Incorrect.
False.

Correct! The log-likelihood function is not concave as a function of $\mu$ and $\Sigma$. However, it is concave with the change of variables $S = \Sigma^{-1}$ and $r = \Sigma^{-1}\mu$.

We can find the maximum-likelihood estimates of $\mu$ and $\Sigma$ by solving a convex optimization problem.

True.

Correct! The trick is to use the variables $S=\Sigma^{-1}$ and $r = \Sigma^{-1}\mu$.
False.

Incorrect.