Suppose \(X_1,\cdots ,X_m\) are independent with \(X_j\sim \operatorname {Binomial}(n_j,p_j)\).
(a) Find the UMVUE’s of \(p_1,\cdots ,p_m\).
(b) Suppose you know that \(p_1=p_2=\cdots =p_m\). Let \(p\) denote this common (but unknown) value. Find the UMVUE of \(p\).
(c) Find the likelihood ratio statistic, \(\lambda \), for \(H_0:p_1=\cdots =p_m\) versus \(H_1:p_i\ne p_j\), for some \(i\) and \(j\).
(d) 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.