## adjustment of bias and coverage for confidence intervals

**M**enéndez, 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.

^{*}**S**ince 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.

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