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/* stdtr.c
*
* Student's t distribution
*
*
*
* SYNOPSIS:
*
* double t, stdtr();
* short k;
*
* y = stdtr( k, t );
*
*
* DESCRIPTION:
*
* Computes the integral from minus infinity to t of the Student
* t distribution with integer k > 0 degrees of freedom:
*
* t
* -
* | |
* - | 2 -(k+1)/2
* | ( (k+1)/2 ) | ( x )
* ---------------------- | ( 1 + --- ) dx
* - | ( k )
* sqrt( k pi ) | ( k/2 ) |
* | |
* -
* -inf.
*
* Relation to incomplete beta integral:
*
* 1 - stdtr(k,t) = 0.5 * incbet( k/2, 1/2, z )
* where
* z = k/(k + t**2).
*
* For t < -2, this is the method of computation. For higher t,
* a direct method is derived from integration by parts.
* Since the function is symmetric about t=0, the area under the
* right tail of the density is found by calling the function
* with -t instead of t.
*
* ACCURACY:
*
* Tested at random 1 <= k <= 25. The "domain" refers to t.
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -100,-2 50000 5.9e-15 1.4e-15
* IEEE -2,100 500000 2.7e-15 4.9e-17
*/
/* stdtri.c
*
* Functional inverse of Student's t distribution
*
*
*
* SYNOPSIS:
*
* double p, t, stdtri();
* int k;
*
* t = stdtri( k, p );
*
*
* DESCRIPTION:
*
* Given probability p, finds the argument t such that stdtr(k,t)
* is equal to p.
*
* ACCURACY:
*
* Tested at random 1 <= k <= 100. The "domain" refers to p:
* Relative error:
* arithmetic domain # trials peak rms
* IEEE .001,.999 25000 5.7e-15 8.0e-16
* IEEE 10^-6,.001 25000 2.0e-12 2.9e-14
*/
/*
Cephes Math Library Release 2.3: March, 1995
Copyright 1984, 1987, 1995 by Stephen L. Moshier
*/
#include "mconf.h"
#include "cephes.h"
extern double PI, MACHEP, MAXNUM;
double stdtr( k, t )
int k;
double t;
{
double x, rk, z, f, tz, p, xsqk;
int j;
if( k <= 0 )
{
mtherr( "stdtr", DOMAIN );
return(0.0);
}
if( t == 0 )
return( 0.5 );
if( t < -2.0 )
{
rk = k;
z = rk / (rk + t * t);
p = 0.5 * incbet( 0.5*rk, 0.5, z );
return( p );
}
/* compute integral from -t to + t */
if( t < 0 )
x = -t;
else
x = t;
rk = k; /* degrees of freedom */
z = 1.0 + ( x * x )/rk;
/* test if k is odd or even */
if( (k & 1) != 0)
{
/* computation for odd k */
xsqk = x/sqrt(rk);
p = atan( xsqk );
if( k > 1 )
{
f = 1.0;
tz = 1.0;
j = 3;
while( (j<=(k-2)) && ( (tz/f) > MACHEP ) )
{
tz *= (j-1)/( z * j );
f += tz;
j += 2;
}
p += f * xsqk/z;
}
p *= 2.0/PI;
}
else
{
/* computation for even k */
f = 1.0;
tz = 1.0;
j = 2;
while( ( j <= (k-2) ) && ( (tz/f) > MACHEP ) )
{
tz *= (j - 1)/( z * j );
f += tz;
j += 2;
}
p = f * x/sqrt(z*rk);
}
/* common exit */
if( t < 0 )
p = -p; /* note destruction of relative accuracy */
p = 0.5 + 0.5 * p;
return(p);
}
double stdtri( k, p )
int k;
double p;
{
double t, rk, z;
int rflg;
if( k <= 0 || p <= 0.0 || p >= 1.0 )
{
mtherr( "stdtri", DOMAIN );
return(0.0);
}
rk = k;
if( p > 0.25 && p < 0.75 )
{
if( p == 0.5 )
return( 0.0 );
z = 1.0 - 2.0 * p;
z = incbi( 0.5, 0.5*rk, fabs(z) );
t = sqrt( rk*z/(1.0-z) );
if( p < 0.5 )
t = -t;
return( t );
}
rflg = -1;
if( p >= 0.5)
{
p = 1.0 - p;
rflg = 1;
}
z = incbi( 0.5*rk, 0.5, 2.0*p );
if( MAXNUM * z < rk )
return(rflg* MAXNUM);
t = sqrt( rk/z - rk );
return( rflg * t );
}
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