[bathyspheric]

Can I ask if I understand your use of `tail` in tail variance? Do you mean tail as in extremes of a distribution, or tails, as in losses. I'm very interested in using MC techniques to model extremes of a distribution, like 1-in-1000 year events from a Weibull distribution, but my (naive) algorithms spend a lot of time in the fat belly of the curve rather than out in the tails, but perhaps (probably) I'm holding it wrong. Is there a way to constrain the MC sampling to the tails of a distribution?

[roenxi]

I meant tail as in the extremes of the distribution. My point was more that once you've assumed that your events follow a Weibull distribution then the opportunities for MCM to add more insight are limited if the task at hand is to predict what the distribution and its parameters are. Obviously if you simulate a Weibull distribution you will get results in line with the info box on the Wikipedia page for the Weibull distribution, and you need to effectively assume parameters to simulate. Simulating a distribution just creates a very error-prone calculator the obscures the assumptions. I've seen people do exactly that thinking it was somehow helping - it is surprisingly easy to accidentally simulate a normal distribution, for example, because if you add a whole bunch of uncorrelated random things together (with identical distributions) then the result is normally distributed. You don't need a simulation to tell you that, and it is easier to mull over without simulation.

If I were using simulation it is because I think that something fishy happens in extreme events (eg, maybe stock returns all starting to become highly correlated in a liquidity crisis, destroying i.i.d assumptions). Or as other commenters mentioned because the thing being simulated has a distribution that is not analytically tractable. But there has to be some phenomena in there that is more complicated than standard distributions or a steady state Markov model, because they are more productive models when they work.

> Is there a way to constrain the MC sampling to the tails of a distribution?

You could sample 100 numbers at a time and drop out the middle 95? The question is maybe not well posed.

[salty_biscuits]

You want to look up about importance sampling. There are lots of ways to do it, the two most common are via scaling or translation.