# [Rd] Density function for non-central t distribution

Claus Ekstroem claus at ekstroem.dk
Wed Jul 16 10:35:50 MEST 2003

```Hi,

I've written some C code for density evaluation of the non-central t distribution. It works
with the R-1.7.1 source code if placed in the src/nmath directory and after appropriate
changes are made to Makefiles, to the dt function in src/library/base/R/distn.R etc.
I haven't read a lot of R source code so it may need some R-ification, but I've tried to use the
dt.c file as a standard.

Use and abuse.

Claus

===> File dnt.c <===

/*
*  AUTHOR
*    Claus Ekstrøm, ekstrom at dina.kvl.dk
*    July 15, 2003.
*
*  This program is free software; you can redistribute it and/or modify
*  the Free Software Foundation; either version 2 of the License, or
*  (at your option) any later version.
*
*  This program is distributed in the hope that it will be useful,
*  but WITHOUT ANY WARRANTY; without even the implied warranty of
*  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
*  GNU General Public License for more details.
*
*  You should have received a copy of the GNU General Public License
*  along with this program; if not, write to the Free Software
*  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307 USA.
*
*
*  NOTE
*
*    Requires the following auxiliary routines:
*
*      lgammafn(x)     - log gamma function
*      pnt(x, df, ncp) - the distribution function for the non-central t distribution
*
*
* DESCRIPTION
*
*    The non-central t density is
*
*         f(x, df, ncp) =
df^(df/2)*exp(-.5*ncp^2)/(sqrt(pi)*gamma(df/2)*(df+x^2)^((df+1)/2))*sum_{k=0}^Infinity gamma((df +
k + df)/2)*ncp^k/prod(1:k)*(2*x^2/(df+x^2))^(k/2)
*
*    The functional relationship
*
*         f(x, df, ncp) = df/x*(F(sqrt((df+2)/df)*x, df+2, ncp) - F(x, df, ncp))
*
*    is used to evaluate the density at x != 0 and
*
*         f(0, df, ncp) = exp(-.5*ncp^2) / (sqrt(pi)*sqrt(df)*gamma(df/2))*gamma((df+1)/2)
*
*    is used for x=0.
*
*    All calculations are done on log-scale to increase stability.
*
*/

#include "nmath.h"
#include "dpq.h"

double dnt(double x, double df, double ncp, int give_log)
{
double u;
#ifdef IEEE_754
if (ISNAN(x) || ISNAN(df))
return x + df;
#endif

// If non-positive df then error
if (df <= 0) ML_ERR_return_NAN;

// If x is infinite then return 0
if(!R_FINITE(x))
return R_D__0;

// If infinite df then the density is identical to a
// normal distribution with mean = ncp
if(!R_FINITE(df))
return dnorm(x, ncp, 1., give_log);

// Consider two cases: x==0 or not
// Do calculations on log scale to stabilize
if (x!=0) {
u = log(df)-log(fabs(x)) + log(fabs(pnt(x*sqrt((df+2)/df), df+2, ncp, 1, 0) - pnt(x, df, ncp,
1, 0)));
}
else {
u = lgammafn((df+1)/2) - lgammafn(df/2) - .5*(log(M_PI) + log(df) + ncp*ncp);
}

return (give_log ? u : exp(u));
}

--
*****************************************
Claus Thorn Ekstrøm <ekstrom at dina.kvl.dk>
Dept of Mathematics and Physics, KVL
Thorvaldsensvej 40
DK-1871 Frederiksberg C
Denmark
Phone:[+45] 3528 2341
Fax:  [+45] 3528 2350

```