I always have trouble with Minimizing MSE problems where MSE stands for the Mean Squared Error. Here is one such problem that gives me trouble:
A random sample of size n is taken from a uniform distribution, $U(0,\theta)$. Find the value $c$ such that an estimate of $\theta$ of the form $c X_{n:n} $ minimizes the mean squared error.
$X_{n:n}$ is the order statistic, for example $X_{1:n}$ is the first order statistic and $X_{n:n}$ is the largest order statistic.
I know to minimize the mean square error you have to set its derivative to $0$ and its general form is $MSE(T) = Var(T) + [b(T)]^2$, where $b(T)$ is the bias function, but how would you find the MSE in this case?