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Ben Bolker

2019-Jun-23 08:39 UTC

[Rd] Calculation of e^{z^2/2} for a normal deviate z

I agree with many the sentiments about the wisdom of computing very
small p-values (although the example below may win some kind of a prize:
I've seen people talking about p-values of the order of 10^(-2000), but
never 10^(-(10^8)) !).  That said, there are a several tricks for
getting more reasonable sums of very small probabilities.  The first is
to scale the p-values by dividing the *largest* of the probabilities,
then do the (p/sum(p)) computation, then multiply the result (I'm sure
this is described/documented somewhere).  More generally, there are
methods for computing sums on the log scale, e.g.

https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.misc.logsumexp.html

 I don't know where this has been implemented in the R ecosystem, but
this sort of computation is the basis of the "Brobdingnag" package for
operating on very large ("Brobdingnagian") and very small
("Lilliputian") numbers.


On 2019-06-21 6:58 p.m., jing hua zhao wrote:> Hi Peter, Rui, Chrstophe and Gabriel,
> 
> Thanks for your inputs --  the use of qnorm(., log=TRUE) is a good point in
line with pnorm with which we devised log(p)  as
> 
> log(2) + pnorm(-abs(z), lower.tail = TRUE, log.p = TRUE)
> 
> that could do really really well for large z compared to Rmpfr. Maybe I am
asking too much since
> 
> z <-20000
>> Rmpfr::format(2*pnorm(mpfr(-abs(z),100),lower.tail=TRUE,log.p=FALSE))
> [1] "1.660579603192917090365313727164e-86858901"
> 
> already gives a rarely seen small p value. I gather I also need a multiple
precision exp() and their sum since exp(z^2/2) is also a Bayes Factor so I  get
log(x_i )/sum_i log(x_i) instead. To this point, I am obliged to clarify, see
https://statgen.github.io/gwas-credible-sets/method/locuszoom-credible-sets.pdf.
> 
> I agree many feel geneticists go to far with small p values which I would
have difficulty to argue againston the other hand it is also expected to see
these in a non-genetic context. For instance the Framingham study was
established in 1948 just got $34m for six years on phenotypewide association
which we would be interesting to see.
> 
> Best wishes,
> 
> 
> Jing Hua
> 
> 
> ________________________________
> From: peter dalgaard <pdalgd at gmail.com>
> Sent: 21 June 2019 16:24
> To: jing hua zhao
> Cc: Rui Barradas; r-devel at r-project.org
> Subject: Re: [Rd] Calculation of e^{z^2/2} for a normal deviate z
> 
> You may want to look into using the log option to qnorm
> 
> e.g., in round figures:
> 
>> log(1e-300)
> [1] -690.7755
>> qnorm(-691, log=TRUE)
> [1] -37.05315
>> exp(37^2/2)
> [1] 1.881797e+297
>> exp(-37^2/2)
> [1] 5.314068e-298
> 
> Notice that floating point representation cuts out at 1e+/-308 or so. If
you want to go outside that range, you may need explicit manipulation of the log
values. qnorm() itself seems quite happy with much smaller values:
> 
>> qnorm(-5000, log=TRUE)
> [1] -99.94475
> 
> -pd
> 
>> On 21 Jun 2019, at 17:11 , jing hua zhao <jinghuazhao at
hotmail.com> wrote:
>>
>> Dear Rui,
>>
>> Thanks for your quick reply -- this allows me to see the bottom of
this. I was hoping we could have a handle of those p in genmoics such as 1e-300
or smaller.
>>
>> Best wishes,
>>
>>
>> Jing Hua
>>
>> ________________________________
>> From: Rui Barradas <ruipbarradas at sapo.pt>
>> Sent: 21 June 2019 15:03
>> To: jing hua zhao; r-devel at r-project.org
>> Subject: Re: [Rd] Calculation of e^{z^2/2} for a normal deviate z
>>
>> Hello,
>>
>> Well, try it:
>>
>> p <- .Machine$double.eps^seq(0.5, 1, by = 0.05)
>> z <- qnorm(p/2)
>>
>> pnorm(z)
>> # [1] 7.450581e-09 1.228888e-09 2.026908e-10 3.343152e-11 5.514145e-12
>> # [6] 9.094947e-13 1.500107e-13 2.474254e-14 4.080996e-15 6.731134e-16
>> #[11] 1.110223e-16
>> p/2
>> # [1] 7.450581e-09 1.228888e-09 2.026908e-10 3.343152e-11 5.514145e-12
>> # [6] 9.094947e-13 1.500107e-13 2.474254e-14 4.080996e-15 6.731134e-16
>> #[11] 1.110223e-16
>>
>> exp(z*z/2)
>> # [1] 9.184907e+06 5.301421e+07 3.073154e+08 1.787931e+09 1.043417e+10
>> # [6] 6.105491e+10 3.580873e+11 2.104460e+12 1.239008e+13 7.306423e+13
>> #[11] 4.314798e+14
>>
>>
>> p is the smallest possible such that 1 + p != 1 and I couldn't find
>> anything to worry about.
>>
>>
>> R version 3.6.0 (2019-04-26)
>> Platform: x86_64-pc-linux-gnu (64-bit)
>> Running under: Ubuntu 19.04
>>
>> Matrix products: default
>> BLAS:   /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.8.0
>> LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.8.0
>>
>> locale:
>>  [1] LC_CTYPE=pt_PT.UTF-8       LC_NUMERIC=C
>>  [3] LC_TIME=pt_PT.UTF-8        LC_COLLATE=pt_PT.UTF-8
>>  [5] LC_MONETARY=pt_PT.UTF-8    LC_MESSAGES=pt_PT.UTF-8
>>  [7] LC_PAPER=pt_PT.UTF-8       LC_NAME=C
>>  [9] LC_ADDRESS=C               LC_TELEPHONE=C
>> [11] LC_MEASUREMENT=pt_PT.UTF-8 LC_IDENTIFICATION=C
>>
>> attached base packages:
>> [1] stats     graphics  grDevices utils     datasets  methods
>> [7] base
>>
>> other attached packages:
>>
>> [many packages loaded]
>>
>>
>> Hope this helps,
>>
>> Rui Barradas
>>
>> ?s 15:24 de 21/06/19, jing hua zhao escreveu:
>>> Dear R-developers,
>>>
>>> I am keen to calculate exp(z*z/2) with z=qnorm(p/2) and p is very
small. I wonder if anyone has experience with this?
>>>
>>> Thanks very much in advance,
>>>
>>>
>>> Jing Hua
>>>
>>>       [[alternative HTML version deleted]]
>>>
>>> ______________________________________________
>>> R-devel at r-project.org mailing list
>>> https://stat.ethz.ch/mailman/listinfo/r-devel
>>>
>>
>>        [[alternative HTML version deleted]]
>>
>> ______________________________________________
>> R-devel at r-project.org mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-devel
> 
> --
> Peter Dalgaard, Professor,
> Center for Statistics, Copenhagen Business School
> Solbjerg Plads 3, 2000 Frederiksberg, Denmark
> Phone: (+45)38153501
> Office: A 4.23
> Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 	[[alternative HTML version deleted]]
> 
> ______________________________________________
> R-devel at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-devel
>

William Dunlap

2019-Jun-23 17:34 UTC

head link

[Rd] Calculation of e^{z^2/2} for a normal deviate z

include/Rmath.h declares a set of 'logspace' functions for use at the C
level.  I don't think there are core R functions that call them.

/* Compute the log of a sum or difference from logs of terms, i.e.,
 *
 *     log (exp (logx) + exp (logy))
 * or  log (exp (logx) - exp (logy))
 *
 * without causing overflows or throwing away too much accuracy:
 */
double  Rf_logspace_add(double logx, double logy);
double  Rf_logspace_sub(double logx, double logy);
double  Rf_logspace_sum(const double *logx, int nx);

Bill Dunlap
TIBCO Software
wdunlap tibco.com


On Sun, Jun 23, 2019 at 1:40 AM Ben Bolker <bbolker at gmail.com> wrote:
>
>   I agree with many the sentiments about the wisdom of computing very
> small p-values (although the example below may win some kind of a prize:
> I've seen people talking about p-values of the order of 10^(-2000), but
> never 10^(-(10^8)) !).  That said, there are a several tricks for
> getting more reasonable sums of very small probabilities.  The first is
> to scale the p-values by dividing the *largest* of the probabilities,
> then do the (p/sum(p)) computation, then multiply the result (I'm sure
> this is described/documented somewhere).  More generally, there are
> methods for computing sums on the log scale, e.g.
>
>
>
https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.misc.logsumexp.html
>
>  I don't know where this has been implemented in the R ecosystem, but
> this sort of computation is the basis of the "Brobdingnag"
package for
> operating on very large ("Brobdingnagian") and very small
> ("Lilliputian") numbers.
>
>
> On 2019-06-21 6:58 p.m., jing hua zhao wrote:
> > Hi Peter, Rui, Chrstophe and Gabriel,
> >
> > Thanks for your inputs --  the use of qnorm(., log=TRUE) is a good
point
> in line with pnorm with which we devised log(p)  as
> >
> > log(2) + pnorm(-abs(z), lower.tail = TRUE, log.p = TRUE)
> >
> > that could do really really well for large z compared to Rmpfr. Maybe
I
> am asking too much since
> >
> > z <-20000
> >>
Rmpfr::format(2*pnorm(mpfr(-abs(z),100),lower.tail=TRUE,log.p=FALSE))
> > [1] "1.660579603192917090365313727164e-86858901"
> >
> > already gives a rarely seen small p value. I gather I also need a
> multiple precision exp() and their sum since exp(z^2/2) is also a Bayes
> Factor so I  get log(x_i )/sum_i log(x_i) instead. To this point, I am
> obliged to clarify, see
>
https://statgen.github.io/gwas-credible-sets/method/locuszoom-credible-sets.pdf
> .
> >
> > I agree many feel geneticists go to far with small p values which I
> would have difficulty to argue againston the other hand it is also expected
> to see these in a non-genetic context. For instance the Framingham study
> was established in 1948 just got $34m for six years on phenotypewide
> association which we would be interesting to see.
> >
> > Best wishes,
> >
> >
> > Jing Hua
> >
> >
> > ________________________________
> > From: peter dalgaard <pdalgd at gmail.com>
> > Sent: 21 June 2019 16:24
> > To: jing hua zhao
> > Cc: Rui Barradas; r-devel at r-project.org
> > Subject: Re: [Rd] Calculation of e^{z^2/2} for a normal deviate z
> >
> > You may want to look into using the log option to qnorm
> >
> > e.g., in round figures:
> >
> >> log(1e-300)
> > [1] -690.7755
> >> qnorm(-691, log=TRUE)
> > [1] -37.05315
> >> exp(37^2/2)
> > [1] 1.881797e+297
> >> exp(-37^2/2)
> > [1] 5.314068e-298
> >
> > Notice that floating point representation cuts out at 1e+/-308 or so.
If
> you want to go outside that range, you may need explicit manipulation of
> the log values. qnorm() itself seems quite happy with much smaller values:
> >
> >> qnorm(-5000, log=TRUE)
> > [1] -99.94475
> >
> > -pd
> >
> >> On 21 Jun 2019, at 17:11 , jing hua zhao <jinghuazhao at
hotmail.com>
> wrote:
> >>
> >> Dear Rui,
> >>
> >> Thanks for your quick reply -- this allows me to see the bottom of
> this. I was hoping we could have a handle of those p in genmoics such as
> 1e-300 or smaller.
> >>
> >> Best wishes,
> >>
> >>
> >> Jing Hua
> >>
> >> ________________________________
> >> From: Rui Barradas <ruipbarradas at sapo.pt>
> >> Sent: 21 June 2019 15:03
> >> To: jing hua zhao; r-devel at r-project.org
> >> Subject: Re: [Rd] Calculation of e^{z^2/2} for a normal deviate z
> >>
> >> Hello,
> >>
> >> Well, try it:
> >>
> >> p <- .Machine$double.eps^seq(0.5, 1, by = 0.05)
> >> z <- qnorm(p/2)
> >>
> >> pnorm(z)
> >> # [1] 7.450581e-09 1.228888e-09 2.026908e-10 3.343152e-11
5.514145e-12
> >> # [6] 9.094947e-13 1.500107e-13 2.474254e-14 4.080996e-15
6.731134e-16
> >> #[11] 1.110223e-16
> >> p/2
> >> # [1] 7.450581e-09 1.228888e-09 2.026908e-10 3.343152e-11
5.514145e-12
> >> # [6] 9.094947e-13 1.500107e-13 2.474254e-14 4.080996e-15
6.731134e-16
> >> #[11] 1.110223e-16
> >>
> >> exp(z*z/2)
> >> # [1] 9.184907e+06 5.301421e+07 3.073154e+08 1.787931e+09
1.043417e+10
> >> # [6] 6.105491e+10 3.580873e+11 2.104460e+12 1.239008e+13
7.306423e+13
> >> #[11] 4.314798e+14
> >>
> >>
> >> p is the smallest possible such that 1 + p != 1 and I couldn't
find
> >> anything to worry about.
> >>
> >>
> >> R version 3.6.0 (2019-04-26)
> >> Platform: x86_64-pc-linux-gnu (64-bit)
> >> Running under: Ubuntu 19.04
> >>
> >> Matrix products: default
> >> BLAS:   /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.8.0
> >> LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.8.0
> >>
> >> locale:
> >>  [1] LC_CTYPE=pt_PT.UTF-8       LC_NUMERIC=C
> >>  [3] LC_TIME=pt_PT.UTF-8        LC_COLLATE=pt_PT.UTF-8
> >>  [5] LC_MONETARY=pt_PT.UTF-8    LC_MESSAGES=pt_PT.UTF-8
> >>  [7] LC_PAPER=pt_PT.UTF-8       LC_NAME=C
> >>  [9] LC_ADDRESS=C               LC_TELEPHONE=C
> >> [11] LC_MEASUREMENT=pt_PT.UTF-8 LC_IDENTIFICATION=C
> >>
> >> attached base packages:
> >> [1] stats     graphics  grDevices utils     datasets  methods
> >> [7] base
> >>
> >> other attached packages:
> >>
> >> [many packages loaded]
> >>
> >>
> >> Hope this helps,
> >>
> >> Rui Barradas
> >>
> >> ?s 15:24 de 21/06/19, jing hua zhao escreveu:
> >>> Dear R-developers,
> >>>
> >>> I am keen to calculate exp(z*z/2) with z=qnorm(p/2) and p is
very
> small. I wonder if anyone has experience with this?
> >>>
> >>> Thanks very much in advance,
> >>>
> >>>
> >>> Jing Hua
> >>>
> >>>       [[alternative HTML version deleted]]
> >>>
> >>> ______________________________________________
> >>> R-devel at r-project.org mailing list
> >>> https://stat.ethz.ch/mailman/listinfo/r-devel
> >>>
> >>
> >>        [[alternative HTML version deleted]]
> >>
> >> ______________________________________________
> >> R-devel at r-project.org mailing list
> >> https://stat.ethz.ch/mailman/listinfo/r-devel
> >
> > --
> > Peter Dalgaard, Professor,
> > Center for Statistics, Copenhagen Business School
> > Solbjerg Plads 3, 2000 Frederiksberg, Denmark
> > Phone: (+45)38153501
> > Office: A 4.23
> > Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com
> >
> >
> >
> >
> >
> >
> >
> >
> >
> >
> >       [[alternative HTML version deleted]]
> >
> > ______________________________________________
> > 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
>
	[[alternative HTML version deleted]]

Martin Maechler

2019-Jun-24 08:37 UTC

head link

[Rd] Calculation of e^{z^2/2} for a normal deviate z

>>>>> William Dunlap via R-devel 
>>>>>     on Sun, 23 Jun 2019 10:34:47 -0700 writes:
>>>>> William Dunlap via R-devel 
>>>>>     on Sun, 23 Jun 2019 10:34:47 -0700 writes:
    > include/Rmath.h declares a set of 'logspace' functions for use
at the C
    > level.  I don't think there are core R functions that call them.

    > /* Compute the log of a sum or difference from logs of terms, i.e.,
    > *
    > *     log (exp (logx) + exp (logy))
    > * or  log (exp (logx) - exp (logy))
    > *
    > * without causing overflows or throwing away too much accuracy:
    > */
    > double  Rf_logspace_add(double logx, double logy);
    > double  Rf_logspace_sub(double logx, double logy);
    > double  Rf_logspace_sum(const double *logx, int nx);

    > Bill Dunlap
    > TIBCO Software
    > wdunlap tibco.com

Yes, indeed, thank you, Bill!

But they *have* been in use by core R functions for a long time
in pgamma, pbeta and related functions.

[and I have had changes in *hyper.c where logspace_add() is used too, 
 for several years now (since 2015) but I no longer know which
 concrete accuracy problem that addresses, so have not yet committed it]


Martin Maechler
ETH Zurich and R Core Team



    > On Sun, Jun 23, 2019 at 1:40 AM Ben Bolker <bbolker at gmail.com>
wrote:

    >> 
    >> I agree with many the sentiments about the wisdom of computing very
    >> small p-values (although the example below may win some kind of a
prize:
    >> I've seen people talking about p-values of the order of
10^(-2000), but
    >> never 10^(-(10^8)) !).  That said, there are a several tricks for
    >> getting more reasonable sums of very small probabilities.  The
first is
    >> to scale the p-values by dividing the *largest* of the
probabilities,
    >> then do the (p/sum(p)) computation, then multiply the result
(I'm sure
    >> this is described/documented somewhere).  More generally, there are
    >> methods for computing sums on the log scale, e.g.
    >> 
    >> 
    >>
https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.misc.logsumexp.html
    >> 
    >> I don't know where this has been implemented in the R
ecosystem, but
    >> this sort of computation is the basis of the
"Brobdingnag" package for
    >> operating on very large ("Brobdingnagian") and very small
    >> ("Lilliputian") numbers.
    >> 
    >> 
    >> On 2019-06-21 6:58 p.m., jing hua zhao wrote:
    >> > Hi Peter, Rui, Chrstophe and Gabriel,
    >> >
    >> > Thanks for your inputs --  the use of qnorm(., log=TRUE) is a
good point
    >> in line with pnorm with which we devised log(p)  as
    >> >
    >> > log(2) + pnorm(-abs(z), lower.tail = TRUE, log.p = TRUE)
    >> >
    >> > that could do really really well for large z compared to
Rmpfr. Maybe I
    >> am asking too much since
    >> >
    >> > z <-20000
    >> >>
Rmpfr::format(2*pnorm(mpfr(-abs(z),100),lower.tail=TRUE,log.p=FALSE))
    >> > [1] "1.660579603192917090365313727164e-86858901"
    >> >
    >> > already gives a rarely seen small p value. I gather I also
need a
    >> multiple precision exp() and their sum since exp(z^2/2) is also a
Bayes
    >> Factor so I  get log(x_i )/sum_i log(x_i) instead. To this point, I
am
    >> obliged to clarify, see
    >>
https://statgen.github.io/gwas-credible-sets/method/locuszoom-credible-sets.pdf
    >> .
    >> >
    >> > I agree many feel geneticists go to far with small p values
which I
    >> would have difficulty to argue againston the other hand it is also
expected
    >> to see these in a non-genetic context. For instance the Framingham
study
    >> was established in 1948 just got $34m for six years on
phenotypewide
    >> association which we would be interesting to see.
    >> >
    >> > Best wishes,
    >> >
    >> >
    >> > Jing Hua
    >> >
    >> >
    >> > ________________________________
    >> > From: peter dalgaard <pdalgd at gmail.com>
    >> > Sent: 21 June 2019 16:24
    >> > To: jing hua zhao
    >> > Cc: Rui Barradas; r-devel at r-project.org
    >> > Subject: Re: [Rd] Calculation of e^{z^2/2} for a normal
deviate z
    >> >
    >> > You may want to look into using the log option to qnorm
    >> >
    >> > e.g., in round figures:
    >> >
    >> >> log(1e-300)
    >> > [1] -690.7755
    >> >> qnorm(-691, log=TRUE)
    >> > [1] -37.05315
    >> >> exp(37^2/2)
    >> > [1] 1.881797e+297
    >> >> exp(-37^2/2)
    >> > [1] 5.314068e-298
    >> >
    >> > Notice that floating point representation cuts out at 1e+/-308
or so. If
    >> you want to go outside that range, you may need explicit
manipulation of
    >> the log values. qnorm() itself seems quite happy with much smaller
values:
    >> >
    >> >> qnorm(-5000, log=TRUE)
    >> > [1] -99.94475
    >> >
    >> > -pd
    >> >
    >> >> On 21 Jun 2019, at 17:11 , jing hua zhao <jinghuazhao
at hotmail.com>
    >> wrote:
    >> >>
    >> >> Dear Rui,
    >> >>
    >> >> Thanks for your quick reply -- this allows me to see the
bottom of
    >> this. I was hoping we could have a handle of those p in genmoics
such as
    >> 1e-300 or smaller.
    >> >>
    >> >> Best wishes,
    >> >>
    >> >>
    >> >> Jing Hua
    >> >>
    >> >> ________________________________
    >> >> From: Rui Barradas <ruipbarradas at sapo.pt>
    >> >> Sent: 21 June 2019 15:03
    >> >> To: jing hua zhao; r-devel at r-project.org
    >> >> Subject: Re: [Rd] Calculation of e^{z^2/2} for a normal
deviate z
    >> >>
    >> >> Hello,
    >> >>
    >> >> Well, try it:
    >> >>
    >> >> p <- .Machine$double.eps^seq(0.5, 1, by = 0.05)
    >> >> z <- qnorm(p/2)
    >> >>
    >> >> pnorm(z)
    >> >> # [1] 7.450581e-09 1.228888e-09 2.026908e-10 3.343152e-11
5.514145e-12
    >> >> # [6] 9.094947e-13 1.500107e-13 2.474254e-14 4.080996e-15
6.731134e-16
    >> >> #[11] 1.110223e-16
    >> >> p/2
    >> >> # [1] 7.450581e-09 1.228888e-09 2.026908e-10 3.343152e-11
5.514145e-12
    >> >> # [6] 9.094947e-13 1.500107e-13 2.474254e-14 4.080996e-15
6.731134e-16
    >> >> #[11] 1.110223e-16
    >> >>
    >> >> exp(z*z/2)
    >> >> # [1] 9.184907e+06 5.301421e+07 3.073154e+08 1.787931e+09
1.043417e+10
    >> >> # [6] 6.105491e+10 3.580873e+11 2.104460e+12 1.239008e+13
7.306423e+13
    >> >> #[11] 4.314798e+14
    >> >>
    >> >>
    >> >> p is the smallest possible such that 1 + p != 1 and I
couldn't find
    >> >> anything to worry about.
    >> >>
    >> >>
    >> >> R version 3.6.0 (2019-04-26)
    >> >> Platform: x86_64-pc-linux-gnu (64-bit)
    >> >> Running under: Ubuntu 19.04
    >> >>
    >> >> Matrix products: default
    >> >> BLAS:   /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.8.0
    >> >> LAPACK:
/usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.8.0
    >> >>
    >> >> locale:
    >> >>  [1] LC_CTYPE=pt_PT.UTF-8       LC_NUMERIC=C
    >> >>  [3] LC_TIME=pt_PT.UTF-8        LC_COLLATE=pt_PT.UTF-8
    >> >>  [5] LC_MONETARY=pt_PT.UTF-8    LC_MESSAGES=pt_PT.UTF-8
    >> >>  [7] LC_PAPER=pt_PT.UTF-8       LC_NAME=C
    >> >>  [9] LC_ADDRESS=C               LC_TELEPHONE=C
    >> >> [11] LC_MEASUREMENT=pt_PT.UTF-8 LC_IDENTIFICATION=C
    >> >>
    >> >> attached base packages:
    >> >> [1] stats     graphics  grDevices utils     datasets 
methods
    >> >> [7] base
    >> >>
    >> >> other attached packages:
    >> >>
    >> >> [many packages loaded]
    >> >>
    >> >>
    >> >> Hope this helps,
    >> >>
    >> >> Rui Barradas
    >> >>
    >> >> ?s 15:24 de 21/06/19, jing hua zhao escreveu:
    >> >>> Dear R-developers,
    >> >>>
    >> >>> I am keen to calculate exp(z*z/2) with z=qnorm(p/2)
and p is very
    >> small. I wonder if anyone has experience with this?
    >> >>>
    >> >>> Thanks very much in advance,
    >> >>>
    >> >>>
    >> >>> Jing Hua
    >> >>>
    >> >>>       [[alternative HTML version deleted]]
    >> >>>
    >> >>> ______________________________________________
    >> >>> R-devel at r-project.org mailing list
    >> >>> https://stat.ethz.ch/mailman/listinfo/r-devel
    >> >>>
    >> >>
    >> >>        [[alternative HTML version deleted]]
    >> >>
    >> >> ______________________________________________
    >> >> R-devel at r-project.org mailing list
    >> >> https://stat.ethz.ch/mailman/listinfo/r-devel
    >> >
    >> > --
    >> > Peter Dalgaard, Professor,
    >> > Center for Statistics, Copenhagen Business School
    >> > Solbjerg Plads 3, 2000 Frederiksberg, Denmark
    >> > Phone: (+45)38153501
    >> > Office: A 4.23
    >> > Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com
    >> >
    >> >
    >> >
    >> >
    >> >
    >> >
    >> >
    >> >
    >> >
    >> >
    >> >       [[alternative HTML version deleted]]
    >> >
    >> > ______________________________________________
    >> > 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
    >> 

    > [[alternative HTML version deleted]]

    > ______________________________________________
    > R-devel at r-project.org mailing list
    > https://stat.ethz.ch/mailman/listinfo/r-devel

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