Inverse cumulative normal distribution

[Dieter]

(01-12-2016 10:16 PM)Pekis Wrote:  I got a somehow gory formula which can be a bit simplified:
(...)
It seems OK with 3 decimals on all range ]0-1[ ... What do you think ?

I wondered how the inverse of these logistic functions (the one you referred to as well as the one I posted) may perform in terms of accuracy. So I calculated Phi(x) for x=0...5 and evaluated the exact inverse for this result. This was compared to the inital x to get the absolute error. Example: Phi(1,2)=0,88483626. The true quantile for this probability is 1,1995157, so the inverse is off by 0,00048.

Here is what I got:

The inverse of the 3rd order approximation you posted works fine for x=0...2,15. This equals a probability of 0,984. On this interval the error of the inverse is within ±0,0012. There is a negative error extremum at x=1,6...1,7, and at x=2,15 the error has risen to the same positive magnitude. For larger x the error grows, at x~3,09 or p=0,999 it has reached about +0,03.

The inverse of the 5th order approximation I suggested is better, but not by much. It works fine for x=0...2,4. This equals a probability of 0,992. On this interval the error of the inverse is within approx. ±0,0003. There is a negative error extremum near x=2, and at x=2,4 the error has risen to the same positive magnitude. For larger x the error grows, at x=2,6 it is as large as the 3rd order approximation. At x=~3,09 or p=0,999 it has reached about +0,006.

So the inverse of these logistic approximations, even if they could be evaluated without much effort, is only useable if one stays away from probabilities close to 0 or 1.

Dieter