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.

While I know the preceding paragraph was crystal clear, let me pile on with an example. Suppose one has a discrete distribution in which there is a 50% chance of 0, a 30% chance of 100,000 and a 20% chance of 800,000. I’ll call this distribution F, just so we can keep track. If one calculates the quantile function of this distribution F, one obtains:

$100000 \text{Boole}[0.5<q\leq 0.8]+800000 \text{Boole}[0.8<q\leq 1.]$, where Boole is a function that takes on a value of 1 when true and 0 otherwise. If one takes the expectation of this quantile function subject to a weighting of 2(1-q) for 0<q<1 one obtains \$53,000.  Note (or just trust me) that this weighting function is a special case of a more general weighting function $\frac{(1-q)^{n-1}}{B(1,n)}$ for n=2, where B is the function Beta.

But now let’s instead take two draws from our distribution function F and look at the distribution of the lowest value.  Its probability density function (PDF) is:

$\begin{cases}\frac{1}{25} & x=800000 \\\frac{21}{100} & x=100000 \\\frac{3}{4} & x=0\end{cases}$

If we take the mean of this PDF, it turns out to be exactly 53,000, the same as the expected quantile.  This sure does not look like a coincidence.  I’ve done a lot experimentation and it appears always to be true that the expectation of a quantile function of a distribution F subject to a weighting function drawn from a beta distribution with one of the parameters 1 and the other parameter an integer N is always equal to the mean of an order distribution of N draws from F where one looks at the lowest value draw.

But precisely because the parameters of a beta distribution do not have to be integers (they just have to be greater than zero), we can extend an order distribution to encompass fractional draws.  To compute the mean of such an order distribution on F with a fractional number of draws D, we compute the expected quantile of the distribution F subject to a weighting distribution BetaDistribution[1,D].

The lovely thing about all this is that it gives an intuitive explanation of risk aversion.  Risk aversion can be thought of as the difference between the mean of some underlying distribution and the mean of its order distribution in which one looks at the lowest value obtained from a number of draws from the distribution.  And risk aversion conceptualized this way can now be a continuum because, thanks to expected quantiles and beta distributions, one can compute the mean of fractional order distributions.

Cool! — particularly since it is sometimes easier to compute the spectral measure than the actual order distribution.