the MC estimator and corresponding interval estimate, for the standard deviation of a normal distribution of unknown mean and variance, are well-known. I would like to find something also for the case of an arbitrary distribution of unknown mean and variance, which is much more often the case when propagating uncertainties through fluid dynamic codes (my case). That's what I came up with so far:
Let X_i be a sequence of real RVs i.i.d with mean mu and variance sigma^2. We form the RVs
They are also i.i.d., clearly, with E[Y_1]=sigma^2 and Var[Y_1]=M_4-sigma^4 where
which I assume to be finite. I introduce the estimator of the mean of the Y_i,
where z_(1-delta/2) is the 1-delta/2 percentile of the standard gaussian distribution. Of course, this expression is useless, because mu and sigma, which are unknown, appear in it. With a bit of hand-waving, I would simply substitute them by the sample mean and the sample variance of the X_i, i.e.