cdf_normal.java

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package statistics;

//Gaussian CDF Taylor approximation
//Code borrowed from http://www1.fpl.fs.fed.us/distributions.html 19/9 2006

/**
*
*This class contains routines to calculate the
*normal cumulative distribution function (CDF) and
*its inverse.
*
*@version .5 --- June 7, 1996
*@version .6 --- January 10, 2001 (normcdf added)
*
*/

public class CDF_Normal extends Object {

/**
*
*This method calculates the normal cdf inverse function.
*<p>
*Let PHI(x) be the normal cdf.  Suppose that Q calculates 
*1.0 - PHI(x), and that QINV calculates QINV(p) for p in (0.0,.5].
*Then for p .le. .5, x = PHIINV(p) = -QINV(p).
*For p .gt. .5, x = PHIINV(p) = QINV(1.0 - p).
*The formula for approximating QINV is taken from Abramowitz and Stegun,
*Handbook of Mathematical Functions, Dover, 9th printing,
*formula 26.2.3, page 933.  The error in x is claimed to
*be less than 4.5e-4 in absolute value.
*
*@param   p    p must lie between 0 and 1.  xnormi returns 
*              the normal cdf inverse evaluated at p.
*
*@author Steve Verrill 
*@version .5 --- June 7, 1996
*
*/

//  FIX: Eventually I should build in a check that p lies in (0,1)

   public static double xnormi(double p) {

      double arg,t,t2,t3,xnum,xden,qinvp,x,pc;

      final double c[] = {2.515517, 
      .802853,
      .010328};

      final double d[] = {1.432788,
      .189269,
      .001308};

      if (p <= .5) {

         arg = -2.0*Math.log(p);
         t = Math.sqrt(arg);
         t2 = t*t;
         t3 = t2*t;

         xnum = c[0] + c[1]*t + c[2]*t2;
         xden = 1.0 + d[0]*t + d[1]*t2 + d[2]*t3;
         qinvp = t - xnum/xden;
         x = -qinvp;

         return x;

      }

      else {

         pc = 1.0 - p;
         arg = -2.0*Math.log(pc);
         t = Math.sqrt(arg);
         t2 = t*t;
         t3 = t2*t;

         xnum = c[0] + c[1]*t + c[2]*t2;
         xden = 1.0 + d[0]*t + d[1]*t2 + d[2]*t3;
         x = t - xnum/xden;

         return x;

      }

   }


/**
*
*This method calculates the normal cumulative distribution function.
*<p>
*It is based upon algorithm 5666 for the error function, from:<p>
*<pre>
*       Hart, J.F. et al, 'Computer Approximations', Wiley 1968
*</pre>
*<p>
*The FORTRAN programmer was Alan Miller.  The documentation
*in the FORTRAN code claims that the function is "accurate
*to 1.e-15."<p>
*Steve Verrill
*translated the FORTRAN code (the March 30, 1986 version)
*into Java.  This translation was performed on January 10, 2001.
*
*@param   z   The method returns the value of the normal
*             cumulative distribution function at z.
*
*@version .5 --- January 10, 2001
*
*/


/*

Here is a copy of the documentation in the FORTRAN code:

	SUBROUTINE NORMP(Z, P, Q, PDF)
C
C	Normal distribution probabilities accurate to 1.e-15.
C	Z = no. of standard deviations from the mean.
C	P, Q = probabilities to the left & right of Z.   P + Q = 1.
C       PDF = the probability density.
C
C       Based upon algorithm 5666 for the error function, from:
C       Hart, J.F. et al, 'Computer Approximations', Wiley 1968
C
C       Programmer: Alan Miller
C
C	Latest revision - 30 March 1986
C

*/

   public static double normp(double z) {

      double zabs;
      double p;
      double expntl,pdf;

      final double p0 = 220.2068679123761;
      final double p1 = 221.2135961699311;
      final double p2 = 112.0792914978709;
      final double p3 = 33.91286607838300;
      final double p4 = 6.373962203531650;
      final double p5 = .7003830644436881;
      final double p6 = .3526249659989109E-01;

      final double q0 = 440.4137358247522;
      final double q1 = 793.8265125199484;
      final double q2 = 637.3336333788311;
      final double q3 = 296.5642487796737;
      final double q4 = 86.78073220294608;
      final double q5 = 16.06417757920695;
      final double q6 = 1.755667163182642;
      final double q7 = .8838834764831844E-1;

      final double cutoff = 7.071;
      final double root2pi = 2.506628274631001;

      zabs = Math.abs(z);

//  |z| > 37

      if (z > 37.0) {

         p = 1.0;

         return p;

      }

      if (z < -37.0) {

         p = 0.0;

         return p;

      }

//  |z| <= 37.

      expntl = Math.exp(-.5*zabs*zabs);

      pdf = expntl/root2pi;

//  |z| < cutoff = 10/sqrt(2).

      if (zabs < cutoff) {

         p = expntl*((((((p6*zabs + p5)*zabs + p4)*zabs + p3)*zabs +
             p2)*zabs + p1)*zabs + p0)/(((((((q7*zabs + q6)*zabs +
             q5)*zabs + q4)*zabs + q3)*zabs + q2)*zabs + q1)*zabs +
             q0);

      } else {

         p = pdf/(zabs + 1.0/(zabs + 2.0/(zabs + 3.0/(zabs + 4.0/
             (zabs + 0.65)))));

      }

      if (z < 0.0) {

         return p;

      } else {

         p = 1.0 - p;

         return p;

      }

   }

}