Let $\theta_1$ be an estimator for $\theta$. We call $\theta_1$ inadmissible if there exists another estimator $\theta_2$ such that $MSE(\theta_2) < MSE(\theta_1)$ for all possible values of $\theta$.
(a) Let $X_1,\cdots, X_n \sim U(0, \theta)$, and define two estimators for $\theta$: Let $\theta_1 = 2\bar{X}$, and let $\theta_2 = \max (X_1,\cdots,X_n)$. Does either estimator prove the other is inadmissible?
(b) Let $Y \sim Expo(\theta)$, and define two estimators for $\theta$: Let $\theta_1 = \frac{1}{X}$, and let $\theta_2 = 4$. Does either estimator prove the other is inadmissible?
What I have so far: For part a, $2\bar{X}$ is an unbiased estimator since the sample mean is an unbiased estimator of the population mean. So the $MSE$ of $2\bar{X}$ would just be the variance of $2\bar{X}$ which is just 2*(variance of sample mean). I am not sure about the $MSE$ of $\theta_2$. How would I figure out its variance and its bias?
For part b, I assume $\theta_2$ is in admissible because it is a constant. How do I compute its mean squared error? I assume it has no variance. What about $\theta_1$? How do I figure out its variance and bias?