Math 6010 • Linear Models

Let \(X_1,X_2,\ldots,X_n\) be independent identically distributed normal \(N(\mu,\sigma^2)\) random variables. Let
\[
U_n=\frac{2}{n(n-1)}
\sum_{1\leq i<j\leq n}I\{|X_i-X_j|>1\}.
\]
\begin{enumerate}
\item Compute \(EU_n\). (10 points)
\item Assume that \(\sigma\) is known. We wish to test \(H_0:\mu=\mu_0\) (\(\mu_0\) is given) against the alternative that \(H_0\) is not true. Can we use \(U_n\) to test \(H_0\) against this alternative? (5 points)
\item Now \(\mu\) and also \(\sigma\) are unknown. We wish to test the null-hypothesis that \(\sigma=\sigma_0\) against the alternative that \(H_0\) is not true. Can we use \(U_n\) now? (5 points)
\end{enumerate}