Parameter estimation for Kumaraswamy distribution

Question

I'm interested in estimating the shape parameters of a Kumaraswamy distribution from sample data. The closest research I can find is Jones' paper from 2009 which analyses a maximum likelihood method, but suggests only generic root-finding for computing the parameter estimates.

Is there more work in this area? Or can somebody suggest even a crude but fast way of estimating the parameters from data?

Reference:

Jones, M.C.: Kumaraswamy’s distribution: A beta-type distribution with some tractability advantages. Statist. Meth. 6, 70-81

Answer 1

The distribution has CDF of $F(x) = 1 - (1-x^a)^b$. By setting $F(x)=1/2$ it is easy to show that the median is $x = (1-2^{-1/b})^{1/a}$. We can obtain a useful equation by setting $x$ equal to the sample median $\hat{\mu}$ and solving: $$\hat{a} = \frac{\log(1-2^{-1/b})}{\log(\hat{\mu})}$$

With this equation you can compute a decent estimate of $a$ given any guess for $b$. To find a good $b$ value you could optimize the ML criterion using a technique similar to golden section search, or use a rootfinding technique to match moments. This is now a 1-dimensional search problem which should make your life a bit easier.