# [Rd] Getting high precision values from qnorm in the tail

Sheldon Maze sheldon.maze at gmail.com
Sun Apr 16 13:30:58 CEST 2017

```Hello All

I am looking for high precision values for the normal distribution in the
tail,(1e-10 and 1 - 1e-10) as the R package that I am using sets any number
which is out of this range to these values and then calls the qnorm and qt
function.

What I have noticed is that the qnorm implementation in R is not symmetric
when looking at the tails. This is quite surprising to me, as it is well
known that this distribution is symmetric, and I have seen implementations
in other languages that are symmetric. I have checked the qt function and
it is also not symmetric in the tails.

Here are the results from the qnorm function:

x       qnorm(x)                qnorm(1-x)              qnorm(1-x) +
qnorm(x)
1e-2    -2.3263478740408408     2.3263478740408408      0.0 (i.e < machine
epsilon)
1e-3    -3.0902323061678132     3.0902323061678132      0.0 (i.e < machine
epsilon)
1e-4    -3.71901648545568       3.7190164854557084
2.8421709430404007e-14
1e-5    -4.2648907939228256     4.2648907939238399
1.014299755297543e-12
1e-10   -6.3613409024040557     6.3613408896974208
-1.2706634855419452e-08

It is quite clear that at a value of x close to 0 or 1, this function
breaks down. Yes, in "normal" use this isn't a problem, but I am looking at
fringe cases and multiplying small probabilities by very large values, in
which case the error (1e-08) becomes a large value.

Note: I have tried this with 1-x and with entering the actual number
0.00001 and 0.99999 and the accuracy issue is still there.

The questions

Firstly, is this a known problem with the qnorm and qt implementations? I
could not find anything in the documentation, the algorithm is supposed to
be accurate 16 digits for p values from 10^-314 as described in the
Algorithm AS 241 paper.

Quote from R doc for qnorm:

"Wichura, M. J. (1988) Algorithm AS 241: The percentage points of the
normal distribution. Applied Statistics, 37, 477–484.

which provides precise results up to about 16 digits."

If the R code implements the 7 digit version, why does it claim 16 digits?
Or is it "accurate" but the original algorithm is not symmetric and wrong?

If R does implement both versions of Algorithm AS 241 can I turn the 16
digit version on?

Or, is there a more accurate version of qnorm in R? Or, another solution to
my problem where I need high precision in the tails for quantile functions.

On a side note, I also have this issue with the qt distribution, at a
similar level of precision, it is not symmetric, nor precise, but I have
not investigated it yet. Also, I've posted this question on stack overflow:
http://stackoverflow.com/questions/43362644/getting-high-precision-values-from-qnorm-in-the-tail

R version:
>version
platform       x86_64-w64-mingw32
arch           x86_64
os             mingw32
system         x86_64, mingw32
status
major          3
minor          3.2
year           2016
month          10
day            31
svn rev        71607
language       R
version.string R version 3.3.2 (2016-10-31)
nickname       Sincere Pumpkin Patch

Kind regards

Sheldon Maze

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```