on approximations of Φ and Φ⁻¹
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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
(with another version replacing 2/π with the squared root of π/8) and
$latex F_2(x)=1/1+exp(-1.5976x(1+0.04417x^2))$
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}}](https://s0.wp.com/latex.php?latex=F_0%5E%7B-1%7D%28p%29+%3D%28-%5Cpi%2F2+%5Clog%5B1-%282p-1%29%5E2%5D%29%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D&bg=000000&%23038;fg=B0B0B0&%23038;s=0&%23038;c=20201002)
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
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