Fractional order distributions using beta-weighted expected quantiles

In an earlier post, I discussed how decisions under uncertainty that could be characterized by a statistical distribution might be evaluated not just using “spectral measures” but also by using an order distribution.  The idea, in a nutshell, was that a risk averse person would evaluate a situation involving uncertainty as if she were making N draws from a distribution and being asked to pick the worst outcome.  “N” becomes a measure of risk aversion.  The higher the “N,” the more likely that one of your draws is going to be really bad. This methodology had the virtue of being perhaps more comprehensible than spectral measures.  A disadvantage, however, was that it permitted only integer measures of risk aversion in which one was restricted to discrete numbers of draws.

What I have now recognized — and I believe this may be what Wolfram Research’s Oleksandr Pavlyk may have been trying to communicate to my less expert self a few weeks ago — is that the mean of an order distribution of some distribution F is — at least often — the expectation of a quantile function of that distribution weighted by a beta distribution in which one of the parameters takes on a value of 1.  And, if this is the case, one can emulate fractional order distributions by letting the other parameter of the beta distribution be something other than an integer.

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