I'm programming a web tool (="I'm a stats ignoramous who drifted here from stackoverflow.com") that allows scientists to enter predictions about the 5-number-summary stats for a variable. Entry is done using the UI-metaphor of a box-plot.
I'd like to allow scientists to visualize their input as a PDF/CDF, but I have to select an underlying distribution.
- I'm looking for a distribution that is "as normal as possible" while still being able to fit well to the 5-number summary pinning down the ~1st, 25th, 50th, 75th and ~99th percentiles.
- I started with the 3-param skew-normal, but it obviously doesn't have enough DOF to perfectly (or even closely) fit to the 5-input parameters
- I'm interpreting 'min' and 'max' as 1st and 99th percentiles. I know this is sketchy, but the numbers entered are speculative predictions (="Don't worry, I'm not screwing up the interpretation of measured data")
- Simplicity is a virtue. Ideally the distribution would be easy to easy-ish to do numerical parameter-estimation with (closed form would be the nicest, ala http://www.johndcook.com/blog/2010/01/31/parameters-from-percentiles/ , but that's pretty much shooting for the moon, doing non-linear optimization or something is fine)
- I've started looking at distributions like GSN/CSN, etc from papers like http://www2.warwick.ac.uk/fac/sci/statistics/crism/research/2012/paper12-08/12-08w.pdf , but I'm not really sure I'm looking down the right family. Maybe skew-normal isn't the best place to start? I've also thought about things like Johnson distribution, which from the little I can find about it seems almost "designed to be fitted".
What distribution(s) should I be looking at?