Math 6010 • Linear Models

Consider the density function

\[f(t) = \frac{1}{C}e^{-t^{4}}, \qquad t \in \mathbb{R},\]

where \(C\) is the constant
\[
C = \int_{-\infty}^{\infty} e^{-t^{4}}\,dt.
\]


Consider the linear model

\[Y_{i} = \mathbf{x}_{i}^{\top} \beta + \epsilon_{i}, 1 \le i \le n,\]


where the \(\epsilon_{i}\) are iid with pdf \(f(t)\). Assume that \(\beta \in \mathbb{R}^{d}\) and \(\sigma^{2} > 0\) are deterministic
but unknown, while \(\mathbf{x}_{i} \in \mathbb{R}^{d}\) are deterministic and known.
\begin{enumerate}

\item
Write down the likelihood function of the observations as a function of \(\beta\) (do not
try to find the maximum likelihood estimator).


\item
Assume temporarily that the \(\epsilon_{i}\) are iid normal with mean zero and variance one,
and write down the likelihood function of the observations as a function of \(\beta\) in that
case.


\item
If you were to solve for the maximum likelihood estimators in the first two parts
would they be the same? (Note you are not being asked to actually solve for the
estimators).

\end{enumerate}