Context: 2nd year university statistics course textbook question
So I had to find two estimators (using method-of-moments and maximum likelihood estimation) of $\theta$ for a random sample $X_1, ..., X_n$ from a population with pmf $f(X=x)=\theta^x(1-\theta)^{1-x}$ for $x=0$ or $x=1$ where $\theta \in [0, 0.5]$ is a model parameter. I recognise this is a Bernoulli distribution.
I found that both methods gave the same estimator $T=\frac{1}{n} \sum^n_{i=1}X_i$ (the sample mean). The next part of the question required me to find the mean squared error of the two estimators. I have a couple questions:
- Since the estimators are the same, does this mean their mean squared erors will be too?
- How should I go about calculating the mean squared error? I know $MSE(T) = Var(T)+[Bias(T)]^2$, but for the $Var(T)$ component I don't know how to calculate $E(T^2)$. Or would it be better to calculate it via $E[(T-\theta)^2]$?
Thanks