Archive for cdf

on approximations of Φ and Φ⁻¹

Posted in Books, Kids, R, Statistics with tags , , , , , , , , , on June 3, 2021 by xi'an

As I was working on a research project with graduate students, I became interested in fast and not necessarily very accurate approximations to the normal cdf Φ and its inverse. Reading through this 2010 paper of Richards et al., using for instance Polya’s

F_0(x) =\frac{1}{2}(1+\sqrt{1-\exp(-2x^2/\pi)})

(with another version replacing 2/π with the squared root of π/8) and


not to mention a rational faction. All of which are more efficient (in R), if barely, than the resident pnorm() function.

      test replications elapsed relative user.self 
3 logistic       100000   0.410    1.000     0.410 
2    polya       100000   0.411    1.002     0.411 
1 resident       100000   0.455    1.110     0.455 

For the inverse cdf, the approximations there are involving numerical inversion except for

F_0^{-1}(p) =(-\pi/2 \log[1-(2p-1)^2])^{\frac{1}{2}}

which proves slightly faster than qnorm()

       test replications elapsed relative user.self 
2 inv-polya       100000   0.401    1.000     0.401
1  resident       100000   0.450    1.000     0.450

open problem

Posted in R, Statistics with tags , , , , , on October 24, 2013 by xi'an

On the plane back from Warwick, I was reading an ABC arXived paper by Umberto Picchini and Julie Forman, “Accelerating inference for diffusions observed with measurement error and large sample sizes using Approximate Bayesian Computation: A case study” and came upon this open problem:

“A closed-form expression for generating percentiles from a fi nite-components Gaussian mixture is unavailable.” (p.5)

which means solving

\alpha\Phi(x)+(1-\alpha)\Phi(\{x-\mu)/\sigma) = \beta

is not possible in closed form. (Of course it could also be argued that the equation Φ(x)=β is unavailable in closed-form ie that the analytic solution x=Φ-1(β) is formal…) While I can think of several numerical approaches, a few minutes with a sheet of paper let me convinced that indeed this is not solvable (hence not an open problem, contrary to the title of the post!).

Just for R practice (and my R course students!), here is a basic R code:


# initial bounds
if (beta<alpha){
if (beta<onmal){

# iterations
for (t in 1:5){

 if ((omB-omb)<prec)
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