Thick tails

Bart Kosko in his book Noise argues that thick-tailed probability distributions such as the Cauchy distribution are common in nature. This is the opposite of what I was taught in college. I remember being told that the Cauchy distribution, a distribution with no mean or variance, is a mathematical curiosity more useful for constructing academic counterexamples than for modeling the real world. Kosko disagrees. He writes

… all too many scientists simply do not know that there are infinitely many different types of bell curves. So they do not look for these bell curves and thus they do not statistically test for them. The deeper problem stems from the pedagogical fact that thick-tailed bell curves get little or no attention in the basic probability texts that we still use to train scientists and engineers. Statistics books for medicine and the social sciences tend to be even worse.

We see thin-tailed distributions everywhere because we don’t think to look for anything else. If we see samples drawn from a thick-tailed distribution, we may throw out the “outliers” before we analyze the data, and then a thin-tailed model fits just fine.

How do you decide what’s an outlier? Two options. You could use your intuition and discard samples that “obviously” don’t belong, or you could use a formal test. But your intuition may implicitly be informed by experience with thin-tailed distributions, and your formal test may also circularly depend on the assumption of a thin-tailed model.