Let $X_i$ be i.i.d. uniform random variables in $[0,\theta]$, for some $\theta>0$ and $M_n = max(X_i)$
I am trying to find the bias of $M_n$ as an estimator of $\theta:$
$$E[M_n] - \theta = $$
Computing $E[M_n] = \frac{n}{n+1}$ I guessed the answer would be $\frac{n}{n+1} - \theta$
But this doesn't seem to be the final answer. What would be the next step? Is there another way?
The Expected value you calculated is wrong.
$$\mathbb{E}[M_n]=\int_0^{\theta}\frac{n}{\theta^n}t^n dt=\frac{n}{n+1}\theta$$
So your estimator is biased but asymptotically correct and its $Bias(M_n)=-\frac{\theta}{n+1}$.
It is asymptotically correct as the Bias goes to zero when $n \rightarrow \infty$
If you are interested in finding the unbiased estimator for $\theta$ it is obviously
$\hat{\theta}=\frac{n+1}{n}M_n$
