nag_gamma_pdf (g01kfc) returns the value of the probability density function (PDF) for the gamma distribution with shape argument and scale argument at a point .
The gamma distribution has PDF
If
then an algorithm based directly on the gamma distribution's PDF is used. For values outside this range, the function is calculated via the Poisson distribution's PDF as described in
Loader (2000) (see
Section 9).
Not applicable.
Not applicable.
Due to the lack of a stable link to
Loader (2000) paper, we give a brief overview of the method, as applied to the Poisson distribution. The Poisson distribution has a continuous mass function given by,
The usual way of computing this quantity would be to take the logarithm and calculate,
Loader introduces an alternative way of expressing
(1) based on the saddle point expansion,
where
, the deviance for the Poisson distribution is given by,
and
For
close to
,
can be evaluated through the series expansion
otherwise
can be evaluated directly. In addition, Loader suggests evaluating
using the Stirling–De Moivre series,
where the error
is given by
Finally
can be evaluated by combining equations
(1)–
(4) to get,
This example prints the value of the gamma distribution PDF at six different points
x with differing
a and
b.