## 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:

See also “Improving ABC for quantile distributions” by Ross McVinish (2010).

5. Chris Drovandi and Tony Pettitt improved upon the Allingham et al paper some time ago now, and extended the models to multivariate quantile distributions (with similar simple call-to-gaussian properties). The paper is in Comp. Stat. Data Anal. in 2011.

Marginally relatedly, Gareth Peters and I applied ABC to g-and-h distributions (milldy similar to g-and-k) in the Journal of Operational Risk in 2006. I haven’t seen anything in this area earlier than this.

• a preprint is available at URL

with motivation for Operational Risk Modeling arising from an influential paper in Risk modeling in Basel II brought out by Dutta and Perry after they studied classes of distributions from this family for a large number of loss data sets, see

In addition there is an excellent relation between g-and-h family of distributions and EVT for those interested in ABC for extremes in the paper of Degen, Embrechts and Lambrigger – see

http://www.actuaires.org/ASTIN/Colloquia/Orlando/Papers/Degen.pdf

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