Math 6010 • Linear Models

Consider the linear model:
\[
Y_{ij}=\beta_i+\epsilon_{ij},\qquad 1\leq i\leq 2,\qquad 1\leq j\leq J.
\]
Suppose \(\epsilon_{ij}\)'s are independent and for some (known) \(a_1\) and \(a_2\), \(\epsilon_{ij}\sim N(0,a_i^2\sigma^2)\).
\begin{parts}
\item Find the U.M.V.U.E.'s of \(\beta_1\) and \(\beta_2\).
\item Suppose you know that for some unknown \(\beta\), \(\beta_1=\beta_2\). Find the U.M.V.U.E. of \(\beta\).
\end{parts}