bdtr.c

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/*                                                     bdtr.c
 *
 *     Binomial distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * int k, n;
 * double p, y, bdtr();
 *
 * y = bdtr( k, n, p );
 *
 * DESCRIPTION:
 *
 * Returns the sum of the terms 0 through k of the Binomial
 * probability density:
 *
 *   k
 *   --  ( n )   j      n-j
 *   >   (   )  p  (1-p)
 *   --  ( j )
 *  j=0
 *
 * The terms are not summed directly; instead the incomplete
 * beta integral is employed, according to the formula
 *
 * y = bdtr( k, n, p ) = incbet( n-k, k+1, 1-p ).
 *
 * The arguments must be positive, with p ranging from 0 to 1.
 *
 * ACCURACY:
 *
 * Tested at random points (a,b,p), with p between 0 and 1.
 *
 *               a,b                     Relative error:
 * arithmetic  domain     # trials      peak         rms
 *  For p between 0.001 and 1:
 *    IEEE     0,100       100000      4.3e-15     2.6e-16
 * See also incbet.c.
 *
 * ERROR MESSAGES:
 *
 *   message         condition      value returned
 * bdtr domain         k < 0            0.0
 *                     n < k
 *                     x < 0, x > 1
 */
/*                                                      bdtrc()
 *
 *      Complemented binomial distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * int k, n;
 * double p, y, bdtrc();
 *
 * y = bdtrc( k, n, p );
 *
 * DESCRIPTION:
 *
 * Returns the sum of the terms k+1 through n of the Binomial
 * probability density:
 *
 *   n
 *   --  ( n )   j      n-j
 *   >   (   )  p  (1-p)
 *   --  ( j )
 *  j=k+1
 *
 * The terms are not summed directly; instead the incomplete
 * beta integral is employed, according to the formula
 *
 * y = bdtrc( k, n, p ) = incbet( k+1, n-k, p ).
 *
 * The arguments must be positive, with p ranging from 0 to 1.
 *
 * ACCURACY:
 *
 * Tested at random points (a,b,p).
 *
 *               a,b                     Relative error:
 * arithmetic  domain     # trials      peak         rms
 *  For p between 0.001 and 1:
 *    IEEE     0,100       100000      6.7e-15     8.2e-16
 *  For p between 0 and .001:
 *    IEEE     0,100       100000      1.5e-13     2.7e-15
 *
 * ERROR MESSAGES:
 *
 *   message         condition      value returned
 * bdtrc domain      x<0, x>1, n<k       0.0
 */
/*                                                      bdtri()
 *
 *      Inverse binomial distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * int k, n;
 * double p, y, bdtri();
 *
 * p = bdtri( k, n, y );
 *
 * DESCRIPTION:
 *
 * Finds the event probability p such that the sum of the
 * terms 0 through k of the Binomial probability density
 * is equal to the given cumulative probability y.
 *
 * This is accomplished using the inverse beta integral
 * function and the relation
 *
 * 1 - p = incbi( n-k, k+1, y ).
 *
 * ACCURACY:
 *
 * Tested at random points (a,b,p).
 *
 *               a,b                     Relative error:
 * arithmetic  domain     # trials      peak         rms
 *  For p between 0.001 and 1:
 *    IEEE     0,100       100000      2.3e-14     6.4e-16
 *    IEEE     0,10000     100000      6.6e-12     1.2e-13
 *  For p between 10^-6 and 0.001:
 *    IEEE     0,100       100000      2.0e-12     1.3e-14
 *    IEEE     0,10000     100000      1.5e-12     3.2e-14
 * See also incbi.c.
 *
 * ERROR MESSAGES:
 *
 *   message         condition      value returned
 * bdtri domain     k < 0, n <= k         0.0
 *                  x < 0, x > 1
 */
 
/*                                                             bdtr() */
 
/*
 * Cephes Math Library Release 2.3:  March, 1995
 * Copyright 1984, 1987, 1995 by Stephen L. Moshier
 */
 
#include "mconf.h"
 
double bdtrc(double k, int n, double p) {
  double dk, dn;
  double fk = floor(k);
 
  if (isnan(p) || isnan(k)) {
    return NAN;
  }
 
  if (p < 0.0 || p > 1.0 || n < fk) {
    sf_error("bdtrc", SF_ERROR_DOMAIN, NULL);
    return NAN;
  }
 
  if (fk < 0) {
    return 1.0;
  }
 
  if (fk == n) {
    return 0.0;
  }
 
  dn = n - fk;
  if (k == 0) {
    if (p < .01)
      dk = -expm1(dn * log1p(-p));
    else
      dk = 1.0 - pow(1.0 - p, dn);
  } else {
    dk = fk + 1;
    dk = incbet(dk, dn, p);
  }
  return dk;
}
 
double bdtr(double k, int n, double p) {
  double dk, dn;
  double fk = floor(k);
 
  if (isnan(p) || isnan(k)) {
    return NAN;
  }
 
  if (p < 0.0 || p > 1.0 || fk < 0 || n < fk) {
    sf_error("bdtr", SF_ERROR_DOMAIN, NULL);
    return NAN;
  }
 
  if (fk == n) return 1.0;
 
  dn = n - fk;
  if (fk == 0) {
    dk = pow(1.0 - p, dn);
  } else {
    dk = fk + 1.;
    dk = incbet(dn, dk, 1.0 - p);
  }
  return dk;
}
 
double bdtri(double k, int n, double y) {
  double p, dn, dk;
  double fk = floor(k);
 
  if (isnan(k)) {
    return NAN;
  }
 
  if (y < 0.0 || y > 1.0 || fk < 0.0 || n <= fk) {
    sf_error("bdtri", SF_ERROR_DOMAIN, NULL);
    return NAN;
  }
 
  dn = n - fk;
 
  if (fk == n) return 1.0;
 
  if (fk == 0) {
    if (y > 0.8) {
      p = -expm1(log1p(y - 1.0) / dn);
    } else {
      p = 1.0 - pow(y, 1.0 / dn);
    }
  } else {
    dk = fk + 1;
    p = incbet(dn, dk, 0.5);
    if (p > 0.5)
      p = incbi(dk, dn, 1.0 - y);
    else
      p = 1.0 - incbi(dn, dk, y);
  }
  return p;
}