Suppose I have some random variable $X$ which only takes on values over some finite region of the real line, and I want to estimate the maximum value of this random variable. Obviously one crude method is to take many measurements, lets say $X_1$, $X_2$, $\ldots, X_n$ (which we'll say are all iid) and to use $X_{max} = \text{max}(X_1, \ldots X_n)$ as my guess, and as long as $n$ is large enough this should be good enough. However, $X_{max}$ is always less than the actual maximum, and I'm wondering if there's any way to modify $X_{max}$ so it gives a guess (still with some uncertainty) which is centred around the actual maximum value, rather than always a little less than it.
Thanks