The Laplace distribution is pointy in the middle and fat in the tails relative to the normal distribution.This post is about a probability distribution that is more pointy in the middle and fatter in the tails.
Here are pictures of the normal and Laplace (a.k.a. double exponential) distributions.
The normal density is proportional to exp(- x2/2) and the Laplace distribution is proportional to exp(-|x|). Near the origin, the normal density looks like 1 – x2/2 and the Laplace density looks like 1 – |x|. And as x gets large, the normal density goes to zero much faster than the Laplace.
Now let’s look at the distribution with density
f(x) = log(1 + 1/x²)
I don’t know a name for this. I asked on Cross Validated whether there was a name for this distribution and no knew of one. The density is related to the bounds on a density presented in this paper. Here’s a plot.
The density is unbounded near the origin, blowing up like -2 log( |x| ) as x approaches 0, and so is more pointed than the Laplace density. As x becomes large, log(1 + x-2) is asymptotically x-2 so the distribution has the same tail behavior as a Cauchy distribution, much heavier tailed than the Laplace density.
Here’s a plot of this new density and the Laplace density together to make the contrast more clear.
As William Huber pointed out in his answer on Cross Validated, this density has a closed-form CDF:
F(x) = 1/2 + (arctan(x) – x log( sin( arctan(x) ) ))/π
The paper mentioned above used a similar density as a Bayesian prior distribution in situations where many observations were expected to be small, though large values were expected as well.