| /* |
| * AUTHOR |
| * Claus Ekstrøm, ekstrom@dina.kvl.dk |
| * July 15, 2003. |
| * |
| * Merge in to R: |
| * Copyright (C) 2003-2015 The R Foundation |
| * |
| * 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, a copy is available at |
| * https://www.R-project.org/Licenses/ |
| * |
| * |
| * NOTE |
| * |
| * Requires the following auxiliary routines: |
| * |
| * lgammafn(x) - log gamma function |
| * pnt(x, df, ncp) - the distribution function for |
| * the non-central t distribution |
| * |
| * |
| * DESCRIPTION |
| * |
| * From Johnson, Kotz and Balakrishnan (1995) [2nd ed.; formula (31.15), p.516], |
| * the non-central t density is |
| * |
| * f(x, df, ncp) = |
| * |
| * exp(-.5*ncp^2) * gamma((df+1)/2) / (sqrt(pi*df)* gamma(df/2)) * (df/(df+x^2))^((df+1)/2) * |
| * sum_{j=0}^Inf gamma((df+j+1)/2)/(factorial(j)* gamma((df+1)/2)) * (x*ncp*sqrt(2)/sqrt(df+x^2))^ j |
| * |
| * |
| * The functional relationship |
| * |
| * f(x, df, ncp) = df/x * |
| * (F(sqrt((df+2)/df)*x, df+2, ncp) - F(x, df, ncp)) |
| * |
| * is used to evaluate the density at x != 0 and |
| * |
| * f(0, df, ncp) = exp(-.5*ncp^2) / |
| * (sqrt(pi)*sqrt(df)*gamma(df/2))*gamma((df+1)/2) |
| * |
| * is used for x=0. |
| * |
| * All calculations are done on log-scale to increase stability. |
| * |
| * FIXME: pnt() is known to be inaccurate in the (very) left tail and for ncp > 38 |
| * ==> use a direct log-space summation formula in that case |
| */ |
| |
| #include "nmath.h" |
| #include "dpq.h" |
| |
| double dnt(double x, double df, double ncp, int give_log) |
| { |
| double u; |
| #ifdef IEEE_754 |
| if (ISNAN(x) || ISNAN(df)) |
| return x + df; |
| #endif |
| |
| /* If non-positive df then error */ |
| if (df <= 0.0) ML_ERR_return_NAN; |
| |
| if(ncp == 0.0) return dt(x, df, give_log); |
| |
| /* If x is infinite then return 0 */ |
| if(!R_FINITE(x)) |
| return R_D__0; |
| |
| /* If infinite df then the density is identical to a |
| normal distribution with mean = ncp. However, the formula |
| loses a lot of accuracy around df=1e9 |
| */ |
| if(!R_FINITE(df) || df > 1e8) |
| return dnorm(x, ncp, 1., give_log); |
| |
| /* Do calculations on log scale to stabilize */ |
| |
| /* Consider two cases: x ~= 0 or not */ |
| if (fabs(x) > sqrt(df * DBL_EPSILON)) { |
| u = log(df) - log(fabs(x)) + |
| log(fabs(pnt(x*sqrt((df+2)/df), df+2, ncp, 1, 0) - |
| pnt(x, df, ncp, 1, 0))); |
| /* FIXME: the above still suffers from cancellation (but not horribly) */ |
| } |
| else { /* x ~= 0 : -> same value as for x = 0 */ |
| u = lgammafn((df+1)/2) - lgammafn(df/2) |
| - (M_LN_SQRT_PI + .5*(log(df) + ncp*ncp)); |
| } |
| |
| return (give_log ? u : exp(u)); |
| } |