Under symmetry, $F^{−1} (0.5) = E(X)$. Compare the sample mean as an estimator of µ to that of the sample median ($F^{−1} (0.5)$) for n sufficiently large, assuming that $X_i$ ∼ Z = N(0, 1), i = 1, . . . , n, iid.
I know that the sample mean is an unbiased estimator of µ, but I'm not sure how to deal with the sample median. It seems the sample median is also unbiased as an estimator of µ, then what should I do to compare them? Can anyone give some hints? Thanks.
Yes here are some hints:
Sample mean is the maximum likelihood estimator if the data at hand follows a Gaussian distribution. On the other hand, sample median is the maximum likelihood estimator of the mean if the data follows a Laplace distribution.
In your question the mean and the median are the same. So both estimators will be able to approach to the true parameter, as long as there are enough data samples, i.e. for large $n$.
The one which has the lowest variance and hence the lowest mean squared error (MSE).
Added: Regarding Batman's comment, one may want to see this answer as well.