Suppose we have N independently drawn samples from an unknown random variable $X$. What is the best way to estimate the expectation of $X$?
For simplicity, we can assume that $X$ only returns values between 0 and 1.
The standard way to estimate the mean is just to just take the average of all the samples. But I got to thinking that this isn't optimal. For example, maybe it's better to take the median-of-averages? I.e. to split the samples into some number of equal-sized buckets, compute the average of each bucket, and then take the median of the averages?
Subquestion: what is the best way to estimate the mean assuming $X$ is a normal random variable with standard deviation 1? Even for this goal I'm not sure which is better: median, average, or something else.
Update: After thinking about @Nick's comment, I think there might be no "best" estimator for the mean. In particular, even for normal distributions, I think that taking the sample mean has advantages and disadvantages over taking the median-of-averages or just the median. Overall, I think median-based methods will give a more spread-out answer, but you'll have better certainty to be within some confidence interval than using mean, but I need to do the math to be sure of this. Maybe there are some good references on the topic?