Inverse cumulative normal distribution

[Dieter]

(01-12-2016 10:16 PM)Pekis Wrote:  Here is what seems a good rough approximation of the inverse cumulative normal distribution (Phi^-1(x)) :

Sorry, but the link does not work for me. Instead of a google.be link, could you provide the direkt URL of the document you refer to?

(01-12-2016 10:16 PM)Pekis Wrote:  It claims a maximum error less than 1.4*10^-4 on all range with this function
Phi(x)=1/(1+e^{-007056x^3-1.5976*x})

Approximations for the Normal integral and its inverse seem to exist since these functions were first defined. I prefer rational functions for this kind of calculation. But if you want something of this logistic type, here is what I sometimes use:

Phi(x) = 1/(1+e^(0,00063*x⁵ – 0,07359*x³ – 1,59544*x)

This approximation was designed for x≥0 and has an absolute error < 1,9 E–5 while the largest relative error is < 2,9 E–5.
With slightly different coefficients this can also be optimized for both an absolute and relative error within 2,2 E–5.
In this case use 0.000611, –0.07349 and –1.59552.

(01-12-2016 10:16 PM)Pekis Wrote:  I simply inverted the function to let Wolfram solve a*y³+b*y-ln(1/x-1)=0 on y (where a=-.07056 and b=-1.5976)

I got a somehow gory formula which can be a bit simplified:
If a=-0.07056 and b=-1.5976 and t(x)=ln(1/x-1)+sqrt(ln²(1/x-1)+4b³/(27a²))
then Phi^-1(x)=(t(x)/(2a))^1/3-(2b³/(27a²t(x)))^1/3

It seems OK with 3 decimals on all range ]0-1[ ... What do you think ?

At least for the inverse (quantile) I would prefer a much simpler rational approximation. Even the good old Hastings version from the Fifties yields an absolute error within 4,5 E–4 and requires just one log, one root and a few multiplications/additions. For most applications the domain is limited, e.g. to probabilities ≥ 1E–99, so that a different set of coefficients will further reduce the error.

If you are interested in simple approximations to the Normal integral – especially in those that can be inverted to get an approximation for the quantile – I'd recommend this paper: "Very Simply Explicitly Invertible Approximations of Normal Cumulative and Normal Quantile Function" by Alessandro Soranzo and Emanuela Epure, published in Applied Mathematical Sciences, Vol. 8, 2014, no. 87. There also is an approximation for Phi(x) with an accuracy better than 2 E–4 which requires just three or four integer (!) constants. ;-)

BTW, this paper also mentions the two-coefficient approximation you gave (cf. Ref. 11). So I assume your source is S.R. Bowling, M.T. Khasawneh, S. Kaewkuekool, B.R. Cho: "A logistic approximation to the cumulative normal distribution", Journal of Industrial Engineering and Management, 2 (2009).

Dieter