Math 6010 • Linear Models

Consider the following model:
\[
y_i=f(x_i)+\epsilon_i,\qquad 1\leq i\leq n,
\]
where
\[
f(t)=a_0+a_1t+\cdots+a_pt^p.
\]
The parameters \(a_0,a_1,\ldots,a_p\) are unknown. The errors \(\epsilon_1,\epsilon_2,\ldots,\epsilon_n\) are independent normal \(N(0,\sigma^2)\) random variables. It is assumed that \(x_i\in[0,1]\) and known. We observe \((y_i,x_i)\), \(1\leq i\leq n\).
\begin{enumerate}
\item Estimate the parameters \(a_0,a_1,\ldots,a_p\) and \(\sigma^2\). (15 points)
\item Get the distribution of the estimator for \(\sigma^2\). (5 points)
\end{enumerate}