Consider probability distribution which is unimodal, symmetric and mean value exists. (Of course due to symmetry mean will coincide with mode (position of the maximum)).


**Question** Is sample mean always the "best" estimate of the mean or may be median, whatever can be better ? 


Situation: assume the "tails" are heavy, (but still mean exists), so "sample-mean" will not be very stable, because big deviations will occur with big probability and hence mean will deviate of the "ideal" value. Median estimate seems to be more stable - if big deviations happens rarely median will not be affected by this. 


So may be one can "cook up" something in the middle between mean and median - just filter out values which are at the ends and take average of the others or something like that...



Estimation of mean of the unimodal symmetric distrubution - is "sample mean" the best estimate ?
First consider a non-bayesian setting in which $X\_1,X\_2,...,X\_n$ are i.i.d. with a [Laplace distribution](http://en.wikipedia.org/wiki/Laplace_distribution) with unknown mean $\mu$. Clearly, the maximum-likelihood estimator here is the sample median and not the sample mean. While in general the ML estimator is non-optimal in the sense that it will not yield the minimum MSE for any value of $\mu$, in the limit $n\to\infty$ it becomes optimal and attains the Cramer-Rao lower bound. We may therefore conclude that in this case and for sufficiently large $n$, the sample median will dominate the sample mean.


Regarding the Bayesian case, let's now assume the $\mu$ is uniformly distributed on $[-1,1]$ and $n=1$. In this case the "sample mean" is simply $X\_1$, which is a linear estimator. Clearly, $E[\mu|X\_1]$ is a non-linear function of $X\_1$ and cannot therefore be the sample mean. However, by simple symmetry considerations one can see that if $X\_i=\mu+\tilde{X}\_i$ where $\tilde{X}\_i$ have zero-mean, and are uncorrelated with each other and with $\mu$, then the sample mean $\frac{1}{n}\sum \_{i=1}^n X\_i$ is the optimal unbiased linear estimator of $\mu$ given $X\_1,...,X\_n$.


