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 [[alternative HTML version deleted]] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel |
rtfm: help('qnorm') identifies arguments you overlooked. 1-x generates
roundoff error. Try the following: x <- 10^(-(1:10)) qx <- qnorm(x) q1x <- qnorm(1-x) qlx <- qnorm(x, lower=FALSE) > cbind(x, qx, q1x, qlx, qx.1x=qx+q1x, qx.lx=qx+qlx) x qx q1x qlx qx.1x qx.lx [1,] 1e-01 -1.281552 1.281552 1.281552 0.000000e+00 0 [2,] 1e-02 -2.326348 2.326348 2.326348 0.000000e+00 0 [3,] 1e-03 -3.090232 3.090232 3.090232 0.000000e+00 0 [4,] 1e-04 -3.719016 3.719016 3.719016 2.842171e-14 0 [5,] 1e-05 -4.264891 4.264891 4.264891 1.014300e-12 0 [6,] 1e-06 -4.753424 4.753424 4.753424 -5.809575e-12 0 [7,] 1e-07 -5.199338 5.199338 5.199338 9.784440e-11 0 [8,] 1e-08 -5.612001 5.612001 5.612001 -8.692842e-10 0 [9,] 1e-09 -5.997807 5.997807 5.997807 4.593952e-09 0 [10,] 1e-10 -6.361341 6.361341 6.361341 -1.270663e-08 0 hope this helps. Spencer Graves On 2017-04-16 6:30 AM, Sheldon Maze wrote: > 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 > > [[alternative HTML version deleted]] > > ______________________________________________ > [hidden email] mailing list > https://stat.ethz.ch/mailman/listinfo/r-devel ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel |
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