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Suppose you want to compute the 25% quantile of some random variable. You can do this by first computing the median, and then computing the median of everything less than the median. You can repeat this process as any times as you want, going recursively in either direction from the last computed median, to compute any value in the quantile function to any desired level of accuracy.

The question is: what happens if you replace the median with the mean above? You get, basically, some kind of "mean version" of the quantile function, or something which fits the metaphor of median:quantile function::mean:________. Does there exist a name for this?

Mike Battaglia
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    It is n ight now, but I guess the following is relevant: https://stats.stackexchange.com/questions/137931/when-would-we-use-tantiles-and-the-medial-rather-than-quantiles-and-the-median/142384#142384 – kjetil b halvorsen Mar 30 '23 at 04:26
  • @kjetilbhalvorsen I was just looking at that the other day, heh. I'm not quite sure if it's the same thing, though. The thing I'm suggesting will lead to some kind of alternative quantile function, and its inverse will be a "cumulative [something] function" of some kind, but I don't it'll be the same as the cumulative expectation function. – Mike Battaglia Mar 30 '23 at 04:34
  • One way to ask this is: For a continuous variable $X$, define $R(0)$ and $R(1)$ as the min and max of the variable; for positive integers $a,b$ with $2a+1<2^b$, define $$R(\frac{2a+1}{2^b})=E\left[X:R(\frac{a}{2^{b-1}})<X<R(\frac{a+1}{2^{b-1}})\right]$$ and define $R(p)$ for any $p\in(0,1)$ by continuity. What is this function? $R$ is boring for a uniform distribution, where it coincides with the quantile function; $R$ will probably be awkward to calculate for the normal and most other distributions; but $R$ might have a nice and interesting form for an exponential or triangular distribution. – Matt F. Apr 05 '23 at 17:21

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