Various Hill Plots
I struggle to interpret Hill plots. Maybe the same is true for you? Here are a bunch of examples to help build intuition. Each example comes with five sets of generated data, to indicate how different they can be due to sampling error.
The Hill plot is sometimes used to estimate the tail exponent of a power law. Although estimating tail exponents accurately is very hard and often ends up being somewhat subjective in the end, I figured it might be useful to look at a few examples of Hill plots to get an intuition for what they can look like, and when one should start to expect odd tail behaviour.
Thin-tailed continuous distributions
We begin with the ever intrusive normal distribution. Its Hill plot resembles the first half of a cycloid or something.
Increasing the variance of the distribution does not change anything about the Hill plot.
Changing its mean does not change the shape of the plot, but it affects the magnitude of the alpha values.
Next, we’ll move on to the exponential distribution. This ought to look a lot like the normal distribution since they have sort of the same type of tail.
And it does! What about the logistic distribution, which we use as a poor man’s normal distribution, but which also looks like it has fatter tails?
Well, that also looks like everything else we’ve seen so far. What about lognormal?
It looks like maybe it doesn’t have as sharp of a point on the right end. And if we really crank up the variance on that one, it starts to look heavy-tailed on a histogram, but does the same thing happen on a Hill plot?
No change, except the magnitude got a whole lot smaller! Let’s look at Student’s t distribution, which at low degrees of freedom should have fatter tails than the normal distribution. But first, high degrees of freedom:
Basically normal. Checks out. Then low degrees of freedom.
This certainly has a blunter right end than the normal distribution! What about with just one degree of freedom?
Oh, yeah, look at that! Stabilises at a tail exponent of one for a while.
Let’s leave these relatively well-behaved continuous distributions aside for now, and check out some odd cases.
Bounded distributions
First up is the uniform distribution.
Very aggressive slope down at the beginning, then slowly leveling out like the others. We can do the Beta distribution with small parameters – this is a kind of mixture of uniform and normal.
Looks a lot like the uniform distribution on the Hill plot, which makes sense given the low parameters: it’s still noticeably bounded on the range 0–1. However, a Beta distribution representing many trials near 50 % ought to be more like a bell curve and not have such a hard cut-off. Does that change its Hill plot?
Yes, this looks more like the normal distribution, albeit with a blunter right end, maybe – sort of like the lognormal distribution.
Discrete distributions
What about discrete distributions? Let’s try Poisson.
Hahah, funny! Turns out this is what Hill plots look like for discrete distributions, like this one with binomial draws:
I’m sure one could make art from this. It’s also reassuring the high-trial count binomial starts to look like a normal distribution, as one would expect.
Heavy-tailed distributions
Now, then, what does the Hill plot look like for actual heavy-tailed distributions? We’ll do a relatively well-known one first: Cauchy.
Okay, stable at a tail exponent of one then drops off. What about a Lévy distribution?
Look at that tail exponent! Let’s also see a real power law, the Pareto distribution.
Oh yeah, fair enough!