Logo Search packages:      
Sourcecode: gretl version File versions  Download package

stdtr.c

/*                                        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 2.8:  June, 2000
  Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*/

#include "mconf.h"

double stdtr (int k, double t)
{
    double x, rk, z, f, tz, p, xsqk;
    int j;

    if (k <= 0) {
      mtherr("stdtr", CEPHES_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 (int k, double p)
{
    double t, rk, z;
    int rflg;

    if (k <= 0 || p <= 0.0 || p >= 1.0) {
      mtherr("stdtri", CEPHES_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;
}

Generated by  Doxygen 1.6.0   Back to index