The MEP doesn't work if you assume you know statistics that you don't. Using a thousand statistics from a data sample should not be done because what you measure from the data sample aren't exactly the statistics from the true distribution.

Right, but what people use the MEP for in practice is to do statistical modeling: one has a data set of outcomes and attempts to build a statistical model of it. So you never know any statistic - even the mean - with absolute confidence.