I have two independent binomial $X_1 =$ binomial $(n,p)$ and $X_2 = $ binomial $(2n,p)$
we assume that $n$ is known but $p$ is an unknown parameter
I want to show that $P_1= 1/3n(X_1+X_2)$ and $P_2= 1/2n(X_1+0.5X_2)$
are both consistent estimators.
I already know that $E(P_1)$ and $E(P_2) = p$ and are both unbiased estimators of p but how can I prove that they are consistent?
I think that I'm doing something wrong as I'm getting the same Variance for both estimators.
Var($P_1)$ and Var$(P_2) = p(1-p)$ which would make them both equally efficient.
Could someone please check my solution?
All you have to show is that the variance of your estimator goes to zero as $n$ goes to infinity. $$\textsf{Var}(P_1) = \textsf{Var}(\frac{1}{3n} (X_1 + X_2)) = \frac{1}{9n^2} [\textsf{Var}(X_1) + \textsf{Var}(X_2) ] $$ Using the variance of a binomial that is $\textsf{Var}(X) = np(1-p)$ $$\textsf{Var}(P_1) = \frac{1}{9n^2} [np(1-p) + 2np(1-p) ] = \frac{3np(1-p)}{9n^2} = \frac{p(1-p)}{3n} \rightarrow 0 $$ as $n \rightarrow \infty$.
More info could be found on this YouTube lecture on minute 00:40:23.