If $X_1,\dots,X_n$ are iid $\mathrm{Binomial}(3,p)$, then the maximum likelihood estimator of $p$ is
$$\hat p = \frac{1}{n}\sum_i X_i$$
Find the bias, variance and MSE of $\hat p$?
We are asked to provide theoretical answers. Would
bias equal $E(\hat p) – p$ ? What about variance and MSE?
I am just looking for some pointers to work this out.
Any help would be greatly appreciated.
Best Answer
Bias is zero as $E(\hat p = \frac{1}{n}\sum_i X_i)=\frac{1}{n}E(\sum_i X_i)=\frac{1}{n}\sum_i E(X_i)= \frac{np}{n}=p$
Hence, $MSE=Var(\hat p)+bias^2=Var(\hat p)+0$
$Var(\hat p)=Var(\frac{1}{n}\sum_i X_i)=\frac{1}{n^2}Var(\sum_i X_i)=\frac{1}{n^2}\sum_i Var(X_i)=\frac{np(1-p)}{n^2}=\frac{p(1-p)}{n}$
therefore $MSE(\hat p)=Var(\hat p)=\frac{p(1-p)}{n}$