For $X_i\sim U[0,a]$ where $i=1,2,\dots,n$ so, $E(X_i)=\dfrac a2$.
Is $a'=\max\{X_1,X_2,\dots,X_n\}$ an unbiased estimator of $a$?
This is what I thought.
Since $a'=\max\{X_1,X_2,\dots,X_n\}=X_k$ such that $X_k\ge X_h$ for any $h$,
$E(a')=E(X_k)=\dfrac a2$.
Therefore $a'$ is not an unbiased estimator.
However, I don't think it is properly solved…
Best Answer
The $\text{cdf}$ of the maximum of $n$ iid random variables is the $n^{th}$ power of the single $\text{cdf}$. In this case, $$\text{cdf}(X)=\left(\frac xa\right)^n.$$
The expectation is
$$\int_0^a x\frac na\left(\frac xa\right)^{n-1}dx=\frac{na^{n+1}}{(n+1)a^n},$$ which is smaller than $a$ (but asymptotically unbiaised for large $n$).
Your reasoning is wrong in that the distribution of $\max\{X_k\}$ is not the same as that of a single $X_k$ (obviously the $\max$ favors the larger values), and the expectations do not match.