Sigmas and Student
I saw something yesterday saying that the Japanese bond market had experienced a six standard deviation move. This brought to mind a post I'd written eight years ago.
All probability statements depend on a model. And if you're probability model says an event had a probability six standard deviations from the mean, it's more likely that your model is wrong than that you've actually seen something that rare. I expand on this idea here.
How likely is it that a sample from a random variable will be six standard deviations from its mean? If you have in mind a normal (Gaussian) distribution, as most people do, then the probability is on the order of 1 chance in 10,000,000. Six sigma events are not common for any distribution, but they're not unheard of for distributions with heavy tails.
LetX be a random variable with a Studentt distribution and degrees of freedom. When is small, i.e. no more than 2, the tails of X are so fat that the standard deviation doesn't exist. As the Studentt distribution approaches the normal distribution. So in some sense this distribution interpolates between fat tails and thin tails.
What is the probability thatX takes on a value more than six standard deviations from its mean at 0, i.e. what does the function
f() = Prob(X > 6)
look like as a function of where ^2 = /( - 2) is the variance of X?
As you'd expect, the limit of f() as is the probability of a six-sigma event for a normal distribution, around 10-7 as mentioned above. Here's a plot of f() for > 3. Notice that the vertical axis is on a log scale, i.e. the probability decreases exponentially.
What you might not expect is that f() isn't monotone. It rises to a maximum value before it decays exponentially. In hindsight this makes sense. As 2+ the variance becomes infinite, and the probability of being infinitely far from the mean is 0. Here's a plot of f() between 2 and 3.
So six sigma probabilities for a Student t distribution rise from 0 up to a maximum of around 10-3 then decrease exponentially, then asymptotically approach a value around 10-7.
Related posts- Converting between nines and sigmas
- When = 30 isn't normal enough
- Fat tails and the t-test
- Beer, wine, and statistics