Math 6010 • Linear Models

Let \(X_1,\ldots,X_n\) be an i.i.d. sample which is uniformly chosen from the interval \((\theta_1,\theta_2)\).
\begin{parts}
\item Prove that there are no one-dimensional (i.e., not vector-valued) sufficient statistics for \(h(\theta_1,\theta_2)\) where \(h\) is a one-to-one measurable function.
\item Is there a one-dimensional (i.e., not vector-valued) sufficient statistics for \(\mu=(\theta_1+\theta_2)/2\)?
\end{parts}