From: Charles C. Berry <cberry_at_tajo.ucsd.edu>

Date: Wed, 10 Oct 2007 11:03:22 -0700

R-devel_at_r-project.org mailing list

https://stat.ethz.ch/mailman/listinfo/r-devel Received on Wed 10 Oct 2007 - 18:11:01 GMT

Date: Wed, 10 Oct 2007 11:03:22 -0700

On Wed, 10 Oct 2007, Duncan Murdoch wrote:

> On 10/10/2007 10:35 AM, skylab.gupta@gmail.com wrote:

*>> Full_Name: Skylab Gupta
**>> Version: R version 2.5.1 (2007-06-27)
**>> OS: Windows XP
**>> Submission from: (NULL) (216.82.144.137)
**>>
**>>
**>> Hello,
**>>
**>> I have been playing around with the statistical distributions in R. I think the
**>> computations for the cumulative distribution function of the students t
**>> distribution in R are not very accurate.
**>>
**>> For instance, the cdf of a students t distribution with 13 degrees of freedom at
**>> 1e-4 is reported in R as "0.5000391350986764"; from Mathematica, it seems the
**>> correct value is "0.50003913510150055", only about 9 accurate digits reported in
**>> R.
**>
**> As Charles Berry told you when this was posted to R-help, it looks as
**> though it is Mathematica that is inaccurate. For example, I would
**> expect this plot to be smooth, and it is not in either R or Mathematica,
**> but R is at least monotone:
**>
**> # the Mathematica values
**> plot(diff(z[80:100]), type='l')
**>
**> The R values
**> plot(diff(pt(1e-4, df=80:100)), type='l')
**>
*

Further, if one truly needs to get highly accurate values for

pt( near.zero, df )

recognize that dt(x, df ) is nearly quadratic around x==0 and dominated by a linear component for x > 0.

So, simple quadrature gets the area under the density for (0, near.zero] quite accurately. One knows that pt(0, df) is exactly 0.5, so this can be added to get the result.

This one point quadrature rule is accurate to better than 3e-14 for every df %in% 1:100 :

really.simple.values <- 0.5 +

sapply( 1:100, function(y) dt( 0.5e-04, y ) * 1e-04 )

Three point Gaussian quadrature (is overkill and) seems accurate up to machine precision.

Chuck

*>>
*

>> I also did the following from within R:

*>>
**>> -------------
**>> df<-seq(1,100,by=1)
**>> y<-pt(1e-4,df)
**>> z<-c(0.50003183098839799,0.50003535533895194,0.50003675525997071,0.50003749999985481,0.50003796066840744,0.50003827327749706,0.50003849914427922,0.50003866990364754,0.50003880349244212,0.50003891083995444,0.50003899897813187,0.50003907263208447,0.50003913510150055,0.50003918874627440,0.50003923531785055,0.50003927612461441,0.50003931217478748,0.50003934425324170,0.50003937297989520,0.50003939886014204,0.50003942229165621,0.50003944360703978,0.50003946308016112,0.50003948094039441,0.50003949738053710,0.50003951256485324,0.50003952663295181,0.50003953969680248,0.50003955185925653,0.50003956322006460,0.50003957385523301,0.50003958382054481,0.50003959318443636,0.50003960200394315,0.50003961032679112,0.50003961818144815,0.50003962562026172,0.50003963266089213,0.50003963934773465,0.50003964569404735,0.50003965173577758,0.50003965749688895,0.50003966298323521,0.50003966823056478,0.50003967322766096,0.50003967801868676,0.50003968260005904,0.50003968700228751,0.50003969121916547,0.
**> 500
**>> 03969526955183,0.50003969915340063,0.50003970290428668,0.50003970650705731,0.50003970997149927,0.50003971332909936,0.50003971654204993,0.50003971964040972,0.50003972264367180,0.50003972553808163,0.50003972835715427,0.50003973106835642,0.50003973370765664,0.50003973624942966,0.50003973868896101,0.50003974107556448,0.50003974338818691,0.50003974563557085,0.50003974781567961,0.50003974993203681,0.50003975199594708,0.50003975399737965,0.50003975593675354,0.50003975782715593,0.50003975966389691,0.50003976145762119,0.50003976321975063,0.50003976489560775,0.50003976655049909,0.50003976818673812,0.50003976975798736,0.50003977127434285,0.50003977277055756,0.50003977423495483,0.50003977566285773,0.50003977705769798,0.50003977841313474,0.50003977975147973,0.50003978102874791,0.50003978230822732,0.50003978356836509,0.50003978477872879,0.50003978596096421,0.50003978713049724,0.50003978827577344,0.50003978935715154,0.50003979045422919,0.50003979153680134,0.50003979256756137,0.500039793
**> 589
**>> 57851,0.50003979462027492)
**>>
**>> plot(df,(y-z)/z, type="s")
**>> -------------
**>>
**>> In the above R code, df contains the 100 integers between 1-100, y contains the
**>> cdf of the students t distribution computed at 1e-4 from R, for all the df
**>> degrees of freedom; and z contains the correct values (to 17 decimal digits) of
**>> the students t distribution cdf at 1e-4 computed from Mathematica; when I plot
**>> the relative errors between the computed values from Mathematica and R, it seems
**>> the relative errors are large; we get only about 10-12 digits of accuracy from R
**>> rather than about 15 digits (all this assuming that the Mathematica computed
**>> values are correct).
**>
**> It seems you are making a bad assumption.
**>
**> Duncan Murdoch
**>
**>
**>
**> This happens for all values close to 0 where the cdf is
**>> evaluated.
**>>
**>> I am working on Windows XP, and I installed a precompiled binary version of R.
**>> The following information might also be useful:
**>>
**>> ---------------
**>>> sessionInfo()
**>> R version 2.5.1 (2007-06-27)
**>> i386-pc-mingw32
**>>
**>> locale:
**>> LC_COLLATE=English_United States.1252;LC_CTYPE=English_United
**>> States.1252;LC_MONETARY=English_United
**>> States.1252;LC_NUMERIC=C;LC_TIME=English_United States.1252
**>>
**>> attached base packages:
**>> [1] "stats" "graphics" "grDevices" "utils" "datasets" "methods"
**>> "base"
**>>
**>>> version
**>> platform i386-pc-mingw32
**>> arch i386
**>> os mingw32
**>> system i386, mingw32
**>> status
**>> major 2
**>> minor 5.1
**>> year 2007
**>> month 06
**>> day 27
**>> svn rev 42083
**>> language R
**>> version.string R version 2.5.1 (2007-06-27)
**>> ---------------
**>>
**>> Is there a reason for this loss of accuracy, or am I missing something here?
**>> Thanks.
**>>
**>> ______________________________________________
**>> R-devel_at_r-project.org mailing list
**>> https://stat.ethz.ch/mailman/listinfo/r-devel
**>
**> ______________________________________________
**> R-devel_at_r-project.org mailing list
**> https://stat.ethz.ch/mailman/listinfo/r-devel
**>
*

Charles C. Berry (858) 534-2098 Dept of Family/Preventive Medicine E mailto:cberry_at_tajo.ucsd.edu UC San Diegohttp://famprevmed.ucsd.edu/faculty/cberry/ La Jolla, San Diego 92093-0901

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