I just stumbled across a distribution that approximates the beta distribution but is easier to work with in some ways. It’s called the Kumaraswamy distribution. Apparently it came out of hydrology. The graph below plots the density of the distribution for various parameters. If you’re familiar with the beta distribution, these curves will look very familiar.
The PDF for the Kumaraswamy distribution K(a, b) is
f(x | a, b) = abxa-1(1 – xa)b-1
and the CDF is
F(x | a, b) = 1 – (1 – xa)b.
The most convenient feature of the Kumaraswamy distribution is that its CDF has a simple form. (The CDF for a beta distribution cannot be reduced to elementary functions unless its parameters are integers.) Also, the CDF is easy to invert. That means you can generate a random sample from a K(a, b) distribution by first generating a uniform random value u and then returning
F-1(u) = (1 – (1 – u)1/b)1/a.
If you’re going to use a Kumaraswamy distribution to approximate a beta distribution, the question immediately arises of how to find parameters to get a good approximation. That is, if you have a beta(α, β) distribution that you want to approximate with a K(a, b) distribution, how do you pick a and b?
My first thought was to match moments. That is, pick a and b so that K(a, b) has the same mean and variance as beta(α, β). That may work well, but it would have to be done numerically.
Since the beta(α, β) density is proportional to xα (1-x)β-1 and the K(a, b) distribution is proportional to xa(1 – xa)b, it seems reasonable to set a = α. But how do you pick b? The modes of the two distributions have simple forms and so you could pick b to match modes:
mode K(a, b) = ((a – 1)/(ab – 1))1/a = mode beta(α, β) = (α – 1)/(α + β – 2).
Update: I experimented with the method above, and it’s OK, but not great. Here’s an example comparing a beta(1/2, 1/2) density with a K(1/2, 2 – √2) density.
Here the K density matches the beta density not at the mode but at the minimum. The blue curve, the curve on top, is the beta density.
Here’s another example, this time comparing a beta(5, 3) density and a K(5, 251/40) density.
Again the beta density is the blue curve, on top at the mode.
Maybe the algorithm I suggested for picking parameters is not very good, but I suspect the optimal parameters are not much better. Rather than saying that the Kumaraswamy distribution approximates the beta distribution, I’d say that the Kumaraswamy distribution is capable of assuming roughly the same shapes as the beta distribution. If the only reason you’re using a beta distribution is to get a certain density shape, the Kumaraswamy distribution would be a reasonable alternative. But if you need to approximate a beta distribution closely, it may not work well enough.