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
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.