Menéndez, Fan, Garthwaite, and Sisson—whom I heard in Adelaide on that subject—posted yesterday a paper on arXiv about correcting the frequentist coverage of default intervals toward their nominal level. Given such an interval [L(x),U(x)], the correction for proper frequentist coverage is done by parametric bootstrap, i.e. by simulating n replicas of the original sample from the pluggin density f(.|θ*) and deriving the empirical cdf of L(y)-θ*. And of U(y)-θ*. Under the assumption of consistency of the estimate θ*, this ensures convergence (in the original sampled size) of the corrected bounds.
Since ABC is based on the idea that pseudo data can be simulated from f(.|θ) for any value of θ, the concept “naturally” applies to ABC outcomes, as illustrated in the paper by a g-and-k noise MA(1) model. (As noted by the authors, there always is some uncertainty with the consistency of the ABC estimator.) However, there are a few caveats:
- ABC usually aims at approximating the posterior distribution (given the summary statistics), of which the credible intervals are an inherent constituent. Hence, attempts at recovering a frequentist coverage seem contradictory with the original purpose of the method. Obviously, if ABC is instead seen as an inference method per se, like indirect inference, this objection does not hold.
- Then, once the (umbilical) link with Bayesian inference is partly severed, there is no particular reason to stick to credible sets for [L(x),U(x)]. A more standard parametric bootstrap approach, based on the bootstrap distribution of θ*, should work as well. This means that a comparison with other frequentist methods like indirect inference could be relevant.
- At last, and this is also noted by the authors, the method may prove extremely expensive. If the bounds L(x) and U(x) are obtained empirically from an ABC sample, a new ABC computation must be associated with each one of the n replicas of the original sample. It would be interesting to compare the actual coverages of this ABC-corrected method with a more direct parametric bootstrap approach.