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Stuart_Armstrong comments on Mean of quantiles - Less Wrong Discussion
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Comments (21)
If we apply this to the Cauchy distribution, your sum is one of the Riemann sums on the way to (aside from a constant factor) the integral from -pi/2 to +pi/2 of tan x dx. This integral diverges because at each endpoint it's like the integral of 1/x, but your procedure is a bit like a Cauchy principal value -- it's like taking the limit of the integral from (-pi/2+epsilon) to (pi/2-epsilon).
So it seems like it might misbehave interestingly for distributions with oddly asymmetrical tails, or with singular behaviour "inside", though "misbehave" is a rather unfair term (you can't really expect it to do well when the mean doesn't exist).
I'm not sure how we could answer question 2; what counts as "effective"? Perhaps an extension of the notion of mean is "effective" if it has nice algebraic properties; e.g., pseudomean(X)+pseudomean(Y) = pseudomean(X+Y) whenever any two of the pseudomeans exist, etc. I suspect that that isn't the case, but I'm not sure why :-).
Interestingly, we can imagine doing the integral of G (the inverse of the CDF) that you define. The Cauchy principal value is like integrating G between x- and x+ such that G(x-)=-y and G(x+)=y, and letting y go to infinity. The averaging I described is like integrating G between x and 1-x and letting x tend to zero.