on approximations of Φ and Φ⁻¹

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

F_2(x)=1/1+\exp(-1.5976x(1+0.04417x^2))

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

One Response to “on approximations of Φ and Φ⁻¹”

  1. […] article was first published on R – Xi'an's Og, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here) […]

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