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

(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

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

simulating Maxwell distribution

A question that came out on X validated a few days ago is how to efficiently simulate from a distribution with density

x²φ(x).

(Obviously this density is already properly normalised since the second moment of the standard Normal  distribution is one.) The first solution that came out (by Jarle Tufto) exploits the fact that this density corresponds to a signed root of a χ²(3) variate. This is a very efficient proposal that requires a Gamma sampler and a random sign sampler. Since the cdf is available in closed form,

Φ(x)-xφ(x),

I ran a comparison with a numerical inversion, but this is much slower. I also tried an accept-reject version based on a Normal proposal with a larger variance, but even when optimising this variance, the running time was about twice as large. While checking Devroye (1986) for any possible if unlikely trick, I came upon this distribution twice (p.119 in an unsolved exercise, p.176 presented as the Maxwell distribution). With the remark that, if

X~x²φ(x),  then  Y=UX~φ(x).

Inverting this result leads to X being distributed as

sign(Y)√(Y²-2log(U)),

which recovers the original χ²(3) solution, if slightly (and mysteriously) increasing the simulation speed.