I am seeking a collaborator to develop a new Stata command that implements CDF interpolation. In return, I can offer coauthorship on an associated paper.
The method, which has already been implemented in R, is described in this paper by me, David Hunter, and Kalie Drown:
https://www.sociologicalscience.com/articles-v4-26-641/

In brief, the method involves
(1) interpolating a monotone spline through approximately 10 points of an empirical cumulative distribution function (CDF). The spline is a smooth estimate of the true CDF.
(2) differentiating the CDF to estimate the probability density function (PDF).
(3) applying numeric integration to functions of the PDF and CDF to estimate summary statistics such as the mean, sd, and Gini.
(4) drawing values at random from the fitted PDF/CDF.

I've taken a first swipe at this, but had difficulties replicating known results in test data. Possibly the spline and numeric integration tools I was using in Stata weren't very good. Possibly I just didn't know what I was doing, or didn't try hard enough. Either way, I suspect there's someone on this list who can do a better job than I can.

If this sounds fun to you, please reply to this note or pm me using the contact form at https://paulvonhippel.com/contact/