We wish to estimate:
var(X.bar) = mp(1 - p)/n.
Where S=X1+X2+...+Xn is a random sample from a binomial (m, p) population
1. Find an unbiased estimator of mp(1 - p)/n.
Note that since E[X1] = mp, X1/m estimates p well, so we may try X1 (m - X1)/m2 as an estimator of p(1 - p), and a constant multiple of X1(m - X1) as an estimator of mp(1 - p)/n.
2. Find the optimal estimator using the Rao-Blackwell Theorem.© BrainMass Inc. brainmass.com March 4, 2021, 7:46 pm ad1c9bdddf
Please see the attached file.
Rao- Blackwell theorem
If is a sufficient statistic for and is unbiased for , then there is a function of namely which is unbiased for and has variance lesser than that of .
Thus the Rao- Blackwell ...
The solution contains an application of Rao-Blackwell theorem to find an optimal estimator for a function of binomial parameter.