Consider the linear model
\[Y_{i} = \mathbf {x}_{i}^{\top } \beta + \epsilon _{i}, 1 \le i \le n,\]
where the \(\epsilon _{i}\) are independent normals with \(\mathbb {E}\epsilon _{i} = 0\) and \(\mathbb {E}\epsilon _{i}^{2} = i\sigma ^{2}\). Here \(\beta \in \mathbb {R}^{d}\) and \(\sigma ^{2} > 0\) are unknown parameters, while the \(\mathbf {x}_{i} \in \mathbb {R}^{d}\) are all known and deterministic. Compute the maximum likelihood estimator of \((\beta , \sigma ^{2} )\).