The variables $X_1,X_2,\ldots X_n$ are i.i.d uniform distributed on $[0,\theta]$. $$T=\max\{X_1,\ldots,X_n\}$$ is the estimate of $\theta$. I need to calculate MSE.
I know that
$$\mathrm{MSE}(T)=\mathbb E[(T-\theta)^2]$$ and so
$$\mathrm{MSE}(T)=\mathrm{Var}(T)-(\mathbb E[T]-\theta)^2$$ but I can't calculate $$\mathbb E[\max\{X_1,X_2,\ldots,X_n\}]$$ and $$\mathrm {Var}(\max\{X_1,X_2,\ldots,X_n\}).$$
Can anyone please help me with this?
My suggestion would be to consider that $X_i / \theta \sim$ uniform$(0, 1)$, and the maximum of an i.i.d. sample of uniform random variables is beta distributed, which you can verify by directly calculating $P(\max( U_1, \ldots , U_n) \leq x) = P(\cap_{i=1}^{n} U_i \leq x) = \prod_{i=1}^{n} P(U_i \leq x) = x^n$. You can use this fact to get both $\text{E}[\max (X_1 , \ldots , X_n)]$ and $\text{Var}[\max (X_1 , \ldots , X_n)]$ without too much trouble using known properties of the beta distribution.