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/* cdf/tdistinv.c
*
* Copyright (C) 2002 Jason H. Stover.
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* 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.
*/
#include <config.h>
#include <math.h>
#include <gsl/gsl_cdf.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_sf_gamma.h>
#include <stdio.h>
static double
inv_cornish_fisher (double z, double nu)
{
double a = 1 / (nu - 0.5);
double b = 48.0 / (a * a);
double cf1 = z * (3 + z * z);
double cf2 = z * (945 + z * z * (360 + z * z * (63 + z * z * 4)));
double y = z - cf1 / b + cf2 / (10 * b * b);
double t = GSL_SIGN (z) * sqrt (nu * expm1 (a * y * y));
return t;
}
double
gsl_cdf_tdist_Pinv (const double P, const double nu)
{
double x, ptail;
if (P == 1.0)
{
return GSL_POSINF;
}
else if (P == 0.0)
{
return GSL_NEGINF;
}
if (nu == 1.0)
{
x = tan (M_PI * (P - 0.5));
}
else if (nu == 2.0)
{
double a = 2 * P - 1;
x = a / sqrt (2 * (1 - a * a));
}
ptail = (P < 0.5) ? P : 1 - P;
if (sqrt (M_PI * nu / 2) * ptail > pow (0.05, nu / 2))
{
double xg = gsl_cdf_ugaussian_Pinv (P);
x = inv_cornish_fisher (xg, nu);
}
else
{
/* Use an asymptotic expansion of the tail of integral */
double beta = gsl_sf_beta (0.5, nu / 2);
if (P < 0.5)
{
x = -sqrt (nu) * pow (beta * nu * P, -1.0 / nu);
}
else
{
x = sqrt (nu) * pow (beta * nu * (1 - P), -1.0 / nu);
}
/* Correct nu -> nu/(1+nu/x^2) in the leading term to account
for higher order terms. This avoids overestimating x, which
makes the iteration unstable due to the rapidly decreasing
tails of the distribution. */
x /= sqrt (1 + nu / (x * x));
}
{
double dP, phi;
unsigned int n = 0;
start:
dP = P - gsl_cdf_tdist_P (x, nu);
phi = gsl_ran_tdist_pdf (x, nu);
if (dP == 0.0 || n++ > 32)
goto end;
{
double lambda = dP / phi;
double step0 = lambda;
double step1 = ((nu + 1) * x / (x * x + nu)) * (lambda * lambda / 4.0);
double step = step0;
if (fabs (step1) < fabs (step0))
{
step += step1;
}
if (P > 0.5 && x + step < 0)
x /= 2;
else if (P < 0.5 && x + step > 0)
x /= 2;
else
x += step;
if (fabs (step) > 1e-10 * fabs (x))
goto start;
}
}
end:
return x;
}
double
gsl_cdf_tdist_Qinv (const double Q, const double nu)
{
double x, qtail;
if (Q == 0.0)
{
return GSL_POSINF;
}
else if (Q == 1.0)
{
return GSL_NEGINF;
}
if (nu == 1.0)
{
x = tan (M_PI * (0.5 - Q));
}
else if (nu == 2.0)
{
double a = 2 * (1 - Q) - 1;
x = a / sqrt (2 * (1 - a * a));
}
qtail = (Q < 0.5) ? Q : 1 - Q;
if (sqrt (M_PI * nu / 2) * qtail > pow (0.05, nu / 2))
{
double xg = gsl_cdf_ugaussian_Qinv (Q);
x = inv_cornish_fisher (xg, nu);
}
else
{
/* Use an asymptotic expansion of the tail of integral */
double beta = gsl_sf_beta (0.5, nu / 2);
if (Q < 0.5)
{
x = sqrt (nu) * pow (beta * nu * Q, -1.0 / nu);
}
else
{
x = -sqrt (nu) * pow (beta * nu * (1 - Q), -1.0 / nu);
}
/* Correct nu -> nu/(1+nu/x^2) in the leading term to account
for higher order terms. This avoids overestimating x, which
makes the iteration unstable due to the rapidly decreasing
tails of the distribution. */
x /= sqrt (1 + nu / (x * x));
}
{
double dQ, phi;
unsigned int n = 0;
start:
dQ = Q - gsl_cdf_tdist_Q (x, nu);
phi = gsl_ran_tdist_pdf (x, nu);
if (dQ == 0.0 || n++ > 32)
goto end;
{
double lambda = - dQ / phi;
double step0 = lambda;
double step1 = ((nu + 1) * x / (x * x + nu)) * (lambda * lambda / 4.0);
double step = step0;
if (fabs (step1) < fabs (step0))
{
step += step1;
}
if (Q < 0.5 && x + step < 0)
x /= 2;
else if (Q > 0.5 && x + step > 0)
x /= 2;
else
x += step;
if (fabs (step) > 1e-10 * fabs (x))
goto start;
}
}
end:
return x;
}
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