I'm wondering if there is a known estimator for expected value that is more efficient than the sample average.
If that is not the case for an arbitrary random variable, then maybe there are examples for special kinds of distributions, like Bernoulli.
Maybe there are some special cases where there is at least an unconstructive proof that there exists an estimator more efficient than sample average? Or that there exists a MOST efficient estimator?
EDIT: a "more efficient" estimator as in an unbiased estimator that has a lower variance than the sample mean. Or alternatively an estimator whose expected value converges to the expected value of the distribution and its variance is asymptotically lower than the variance of sample mean as the sample size tends to infinity.