Tick Density of Cumulative Plots

(3) Numerical calculation of erf-1(x)

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<Numerical formulae>

NIST(National Institute of Science and Technology) gives in its website:

  1. Series expansion with respect to t=(x/2)√π (x < x0)
  2. Asymptotic expansion of erfc-1(x') (x'=1-x,  x0< x < 1)
    where u and a are functions of x'
We adopted x0=0.98 so that the two are connected to within 1%. The solid curve is erf-1(x) and the broken curve erf(x). 		<Result>
<Acklam's formula>

Let the cumulative distribution of x be ψ(x), where the average=0 and standard deviation=1. Ackman obtained a fractional approximation to ψ-1(x) (here). With the formula one can calculate erf-1(x) from the formula in the right.

9-18-2023, S. Hayashi