Given $n$ i.i.d. variables $X\_1$ to $X\_n$ with an unknown probability distribution, the sample average is an unbiased estimator for the mean of the distribution. Is there some non-trivial probability distribution for which $\min(X\_1,\ldots,X\_n)$ is an unbiased estimator? (Non-trivial meaning the variables can have more than one potential value).



Is the min function ever an unbiased estimator for the mean?
No. The minimum as always smaller than or equal to the arithmetic mean, and is strictly smaller with positive probability (i.e., when not all the $X\_i$ have the same value). Hence its expected value is strictly smaller than that of the mean.


