## Quantile distributions

Kerrie Mengersen, who is visiting CREST and Dauphine this month, showed me a 2009 paper she had published in Statistics and Computing along with D. Allingham and R. King on an application of ABC to quantile distributions. Those distributions are defined by a closed-form quantile function, which makes them easy to simulate by a simple uniform inversion, and a mostly unavailable density function, which makes any approach but ABC difficult or at least costly to implement. For instance, the g-and-k distribution is given by

$Q(u;A,B,g,k) = \qquad\qquad\qquad$

$\qquad A + B\left[1+c\dfrac{1-\exp\{-g\Phi(u)\}}{1+\exp\{-g\Phi(u)\}}\right]\{1+\Phi(u)^2\}^k\Phi(u)$

hence can be simulated by a single call to a normal simulation. This is therefore a good benchmark for realistic albeit simple examples to use in ABC calibration and we are currently experimenting with it.

### 7 Responses to “Quantile distributions”

1. […] Métodos Bayesianos 11 (no major difference with the slides from Zürich, hey!, except for the quantile distribution example] […]

2. […] book indeed does not aim at fitting standard distributions but instead at promoting a class of quantile distributions, the generalised lambda distributions (GLDs), whose quantile function is a location-scale transform […]

3. Dennis Prangle Says:

I’ve been tidying up some code for the g-and-k distribution from my thesis and posted it as an R package here. Any comments welcome!

• Thank you, Dennis, this is a great benchmark, so the package should be most useful!

4. Chris Drovandi Says: