Math 6010 • Linear Models

Let \(\epsilon_{1}, \ldots, \epsilon_{n}\) and \(\eta_{1}, \ldots, \eta_{m}\) be independent normal random variables with \(\mathbb{E}\epsilon_{i} = \mathbb{E}\eta_{j} = 0\)
and \(\mathbb{E}\epsilon_{i}^{2} = 1, \mathbb{E}\eta_{j}^{2} = 4\). We have the models

\[X_{i} = \beta_{1} + \beta_{2} x_{i} + \epsilon_{i}, 1 \le i \le n,\]

\[Y_{i} = \beta_{1} + \beta_{3} y_{i} + \eta_{i}, 1 \le i \le m,\]

where the \(x_{i}, y_{i}\) are given and non-random.
\begin{enumerate}

\item
Find the maximum likelihood estimator for \((\beta_{1}, \beta_{2}, \beta_{3} )\).


\item
Compute the distribution of the estimator.

\end{enumerate}