We are going to play around with a mixture distribution. A wider distribution is “polluting” a narrower distribution. We will show that mean absolute deviation is a more efficient estimator of dispersion than standard deviation.
Here’s a useful little R mixture distribution function. I give you the code, show how it works, and give some examples.
In the risk business one is not so much concerned with the probability of loss as the expected size of the loss. We explore this idea for various probability distributions by looking at the expected loss given that the loss exceeds a certain threshold. We see for fat tails that the expected loss exceeds the threshold itself.
Resolving a discrepancy between my results and Taleb’s in Taleb’s favor, naturally! I had made an assumption that was not implied in his narrative.
In the real world we generally have only the data and no information about what generated the data. We can go far astray if we make the wrong assumptions.