@jitb  - it might be a mixture, in which case the bootstrap confidence bound is more likely to provide proper coverage.  However, on the log scale, your values range from 0 to about 3.5 with a probable peak around 1.  So a mean on the original scale of 2.7 or thereabouts makes sense.   However, I sense something interesting here.  It looks like your raw values are bounded away from 0.  Have you considered a gamma distribution for the values?  It has a closed form mean and variance (small bias involved compared to ML estimators).  And there is a compound gamma distribution (also with closed form estimators) that is essentially a mixture of two gamma distributions having the same mean but differing variance. PROC FMM on the 1000 simulated values with 2 gamma components compared to a single component by AIC sounds like a good approach.   In any case, that bootstrap mean is still likely to be your best estimator of central tendency..  Using the least weighted average difference should approximate the median, so you could check the 50th percentile value of the bootstrap sample against it.  I worry that the "best" may be way out toward one or the other tails. ... View more

If your mediator looks like it may have a gamma distribution, consider a close approximation such as a lognormal.  To do this, take the natural log of the mediator variable - this will have a normal distribution of residuals in the log space and has a default identity link in the other PROCs that address alternatives to the normal distribution.  It then becomes even more important that you consider how you report the mediating effect.   SteveDenham ... View more

1.96 is fine for a large lognormal population, so long as you are doing calculations in the log space.  Confidence bounds on the original scale could be obtained by exponentiating those obtained using the 1.96 factor on the log space bounds. This is because the variance of the lognormal distribution is not assumed to be a function of the mean, so the logs of the values are assumed to follow a Gaussian distribution.  For analysis of variance purposes, this means that the residuals in the log space are normally distributed.   SteveDenham ... View more

You have what you need for a bootstrap estimate of the mean and confidence interval. I wouldn't choose any "best' simulation as that is going to be strictly a function of the random values used to generate your time series.  Instead, your best predictor is simply the mean of the new point across the 1000 simulations, and the confidence bounds would be as @PaigeMiller pointed out - the 2.5th percentile and the 97.5th percentile.  You can get all of these with one call to PROC MEANS.   SteveDenham ... View more