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).
Loader C (2000) Fast and accurate computation of binomial probabilities (not yet published)
On entry: , the value at which the PDF is to be evaluated.
On entry: , the shape argument of the gamma distribution.
On entry: , the scale argument of the gamma distribution.
– NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).
Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 126.96.36.199 in How to Use the NAG Library and its Documentation for further information.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
Computation abandoned owing to overflow due to extreme parameter values.
On entry, .
On entry, .
Parallelism and Performance
nag_gamma_pdf (g01kfc) is not threaded in any implementation.
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,
For large and , and are very large, of the same order of magnitude and when calculated have rounding errors. The subtraction of these two terms can therefore result in a number, many orders of magnitude smaller and hence we lose accuracy due to subtraction errors. For example for and , and . But calculated with the method shown later we have . The difference between these two results suggests a loss of about 7 significant figures of precision.
Loader introduces an alternative way of expressing (1) based on the saddle point expansion,
where , the deviance for the Poisson distribution is given by,
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.