I can't seem to find a three-parameter t-distribution in numpy/scipy and I know JAGS has one. It turns out it comes from the "jrmath" library which is a part of R.

And from modules/bugs/distributions/DT.cc:

Which gets you the three-parameter version, and suggests that the following NumPy code should work nicely:

...and in fact generating a large sample from this, as well as from R, followed by a quantile-quantile plot gives some confirmation that this works.

This could be considered the historical-materialist approach to coding statistical functions. I'm sure there is a reference text for this sort of thing somewhere but it's not the sort of thing which jumps to mind for a biologist.

It helps to know that

`/*`

* Mathlib : A C Library of Special Functions

* Copyright (C) 1998 Ross Ihaka

* Copyright (C) 2000-2008 The R Development Core Team

*

* 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

* http://www.r-project.org/Licenses/

*

* DESCRIPTION

*

* Pseudo-random variates from a t distribution.

*

* NOTES

*

* This function calls rchisq and rnorm to do the real work.

*/

#include "nmath.h"

double rt(double df, RNG *rng)

{

if (ISNAN(df) || df <= 0.0) ML_ERR_return_NAN;

if(!R_FINITE(df))

return norm_rand(rng);

else {

/* Some compilers (including MW6) evaluated this from right to left

return norm_rand(rng) / sqrt(rchisq(df) / df); */

double num = norm_rand(rng);

return num / sqrt(rchisq(df, rng) / df);

}

}

And from modules/bugs/distributions/DT.cc:

`double DT::r(vector` const &par, RNG *rng) const

{

return rt(DF(par), rng) / sqrt(TAU(par)) + MU(par);

}

Which gets you the three-parameter version, and suggests that the following NumPy code should work nicely:

def student_t_3(loc, df, scale, size):

tau = 1/(scale**2)

t_draw = np.random.standard_t(df = df, size = size)

scaled_t_draw = t_draw / np.sqrt(tau)

return(scaled_t_draw + loc)

...and in fact generating a large sample from this, as well as from R, followed by a quantile-quantile plot gives some confirmation that this works.

This could be considered the historical-materialist approach to coding statistical functions. I'm sure there is a reference text for this sort of thing somewhere but it's not the sort of thing which jumps to mind for a biologist.

It helps to know that

*numpy.random*is coded expecting that everything is parameterized by scale parameters, whereas JAGS is coded expecting that (almost?) everything is parameterized with rate parameters.
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