Math 6010 • Linear Models

Consider the linear model: \(Y_i=\beta+\epsilon_i\), \(1\leq i\leq n\). Here, \(\epsilon_i\)'s are i.i.d. \(N(0,\sigma^2)\), and \(\sigma\) and \(\beta\) are unknown.
\begin{parts}
\item Find the least squares estimator of \(\beta\);
\item Find the UMVU estimator for \(\beta\) (prove the optimality);
\item Find the UMP test for \(H_0:\beta=\beta_0\) versus \(H_1:\beta=\beta_1\);
\item Discuss--without proofs--how to find the UMP test for \(H_0:\beta>\beta_0\) vs. \(H_1:\beta\leq\beta_0\) from tests of the form in part (c) above.
\end{parts}