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($X_1$,…,$X_n$) is an unbiased estimator? (Non-trivial meaning the variables can have more than one potential value).
[Math] Is the min function ever an unbiased estimator for the mean
estimation-theorypr.probabilityst.statistics
Best Answer
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.