Math 6010 • Linear Models

Suppose \(X_1,\ldots,X_m\) are independent with \(X_j\sim \operatorname{BIN}(n_j,p_j)\).
\begin{parts}
\item Find the UMVUE's of \(p_1,\ldots,p_m\).
\item Suppose you know that \(p_1=p_2=\cdots=p_m\). Let \(p\) denote this common (but unknown) value. Find the UMVUE of \(p\).
\item Find the likelihood ratio statistic, \(\lambda\), for \(H_0:p_1=\cdots=p_m\) versus \(H_1:p_i\neq p_j\), for some \(i\) and \(j\).
\item It can be shown that \(2\log\lambda\) is approximately the same as the usual \(\chi^2\) statistics. Using this fact, find an approximate test for \(H_0\) vs \(H_1\) above.
\end{parts}