I have the following exercise, assuming $X_1,...,X_n$ are i.i.d $Exp(\lambda)$,$\lambda>0$, consider the following estimators for the parameter $\mu=\frac{1}{\lambda}$: $$ \hat{\mu_1}=n*min(X_1,...,X_n) $$ $$ \hat{\mu_2}=\frac{1}{n-1}\sum_{i=1}^nX_i $$ Compute the expectation for both parameters, are they unbiased? Also show that, $$ MSE(\hat{\mu_1})=\mu^2 \quad\textbf{and} \quad MSE(\hat{\mu_2})=\frac{n+1}{(n-1)^2}\mu^2 $$
I calculated the expected value for $\hat{\mu_2}=\frac{n}{n-1}\frac{1}{\lambda}$ and got the stated MSE. The second estimator is unbiased and i'm sure the first one also. My problem lies within the first estimator, i have no clue how to calculate the expected value and the variance in order to calculate the MSE. I obtained $$ E[\hat{\mu_1}]=\frac{n}{\lambda} \quad \textbf{and} \quad Var[\hat{\mu_1}]=\frac{n^2}{\lambda^2} $$ but this gives me $MSE(\hat{\mu_1})=\frac{n^2}{\lambda^2}+\frac{(n-1)^2}{\lambda^2}$ so it's obviously wrong. How do i calculate the expected value and variance for $\hat{\mu_1}$? I never encountered before an estimator involving the minimum of a set of random variables. Thanks in advance.