Glad to hear this was of help, Tom!

]]>Since we are in a one dimensional setting and that S is the minimal statistic, it sounds all the same to me. If I change my sample in such a way that S is turned into an arbitrary new value s and the median is unchanged, I cannot find a function f: s -> f(s) that gives the median…

]]>Dan: if you use solely the mean or the median to separate between log-normal and gamma or even the pair of them, the Bayes factor will not ring twice (i.e., will be inconsistent). On the opposite, if you use a collection of quantiles, including the median, you should achieve BF consistency. Now, an interesting question is about which statistic provides the most efficient BF convergence? I suspect, based on P. Fearnhead’s and D. Prangle’s work that it may be the BF itself…

]]>I wonder what this does to the asymptotics. Let’s say you were trying to choose between a log-normal and a gamma model. These both exponential family distributions, so the BF computed from ABC should be consistent. But if we replaced the means in the sufficient statistics with medians, would we still be consistent. I’d hope so, because it’s a more robust measure of identical information, but maybe the robustness of the extremes wouldn’t allow proper discrimination if the distributions differ only in the tails.

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