Is $\Phi^{-1}(\frac{n}{n+1})$ an unbiased estimator of the maximum of $n$ draws from the probability distribution of $\Phi$? Consider a continuous probability distribution $\mathcal{D}$ on $\mathbb{R}$ with cdf $\Phi$ and let $X_1,\dots,X_n\sim\mathcal{D}$ be iid random variables.
Then I have the following conjecture: $$\mathbb{E}\left(\max_iX_i\right)=\Phi^{-1}\left(\frac{n}{n+1}\right)$$
Can someone tell me whether this is true indeed or not?
I conducted several Monte Carlo Simulations and found that it seems to hold for all continuous distributions, but I have no idea how to prove this rigorously myself and I also could not find a relevant theorem.
 A: With $n=1$, your question becomes "is the mean equal to the median?"  The answer is "not necessarily, no," and there's no reason it would become true for larger values of $n$.
A: The intuition you have this is due to the following fact:

*

*As $X$ is continuous, we know that $X \sim \Phi^{-1}(U)$ where $U \sim \text{Uniform}(0, 1)$


*$\Phi^{-1}$ is monotonic increasing (in fact strictly when $X$ is continuous) and thus
$$ \max_i X_i \sim \max_i\Phi^{-1}(U_i) = \Phi^{-1}\left(\max_i U_i\right) $$


*$\displaystyle \max_i U_i$ has a Beta distribution with the expectation
$$ E\left[\max_i U_i\right] = \frac {n} {n+1}  $$
So combining all together, taking expectation from the result 2, we have
$$ E\left[\max_i X_i\right] = E\left[\Phi^{-1}\left(\max_i U_i\right)\right]$$
It is tempted to move the expectation inside $\Phi^{-1}$; However we know that in general they are not equal due to Jensen's inequality:
$$ E\left[\Phi^{-1}\left(\max_i U_i\right)\right] \neq 
\Phi^{-1}\left(E\left[\max_i U_i\right]\right) = 
\Phi^{-1}\left(\frac {n} {n+1}\right)$$
Equality holds only when $\Phi^{-1}$ is linear, i.e. back to the uniform distribution case.
