NAG CL Interface
g01kfc (pdf_​gamma)

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1 Purpose

g01kfc returns the value of the probability density function (PDF) for the gamma distribution with shape parameter α and scale parameter β at a point x.

2 Specification

#include <nag.h>
double  g01kfc (double x, double a, double b, NagError *fail)
The function may be called by the names: g01kfc, nag_stat_pdf_gamma or nag_gamma_pdf.

3 Description

The gamma distribution has PDF
f(x)= 1βαΓ(α) xα-1e-x/β if ​x0;  α,β>0 f(x)=0 otherwise.  
If 0.01x,α,β100 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).

4 References

Loader C (2000) Fast and accurate computation of binomial probabilities (not yet published)

5 Arguments

1: x double Input
On entry: x, the value at which the PDF is to be evaluated.
2: a double Input
On entry: α, the shape parameter of the gamma distribution.
Constraint: a>0.0.
3: b double Input
On entry: β, the scale parameter of the gamma distribution.
  • b>0.0;
  • xb<1nag_real_safe_small_number().
4: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

If fail.code= NE_NOERROR, then g01kfc returns 0.0.
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
Computation abandoned owing to overflow due to extreme parameter values.
On entry, a=value.
Constraint: a>0.0.
On entry, b=value.
Constraint: b>0.0.

7 Accuracy

Not applicable.

8 Parallelism and Performance

g01kfc is not threaded in any implementation.

9 Further Comments

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,
p(x;λ) = λx x! e-λ . (1)
The usual way of computing this quantity would be to take the logarithm and calculate,
log(x;λ) = x logλ - log(x!) - λ .  
For large x and λ, xlogλ and log(x!) 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 x=2×106 and λ=2×106, log(x!)2.7×107 and log(p(x;λ))=-8.17326744645834. But calculated with the method shown later we have log(p(x;λ))=-8.1732674441334492. 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,
log(p(x;λ)) = log(p(x;x)) - D(x;λ) , (2)
where D(x;λ), the deviance for the Poisson distribution is given by,
D(x;λ) = log(p(x;x)) - log(p(x;λ)) , = λ D0 ( x λ ) , (3)
D0 (ε) = ε logε + 1 - ε .  
For ε close to 1, D0(ε) can be evaluated through the series expansion
λ D0 ( x λ ) = (x-λ) 2 x+λ + 2x j=1 v 2j+1 2j+1 ,  where ​ v = x-λ x+λ ,  
otherwise D0(ε) can be evaluated directly. In addition, Loader suggests evaluating log(x!) using the Stirling–De Moivre series,
log(x!) = 12 log (2πx) + x log(x) -x + δ(x) , (4)
where the error δ(x) is given by
δ(x) = 112x - 1 360x3 + 1 1260x5 + O (x-7) .  
Finally log(p(x;λ)) can be evaluated by combining equations (1)(4) to get,
p (x;λ) = 1 2πx e - δ(x) - λ D0 (x/λ) .  

10 Example

This example prints the value of the gamma distribution PDF at six different points x with differing a and b.

10.1 Program Text

Program Text (g01kfce.c)

10.2 Program Data

Program Data (g01kfce.d)

10.3 Program Results

Program Results (g01kfce.r)
GnuplotProduced by GNUPLOT 4.6 patchlevel 3 0 0.05 0.1 0.15 0.2 0.25 0.3 0 1 2 3 4 5 6 7 8 9 10 y x Example Program Plots of the Gamma Distribution α=2, β=2 α=9, β=0.5 gnuplot_plot_1 gnuplot_plot_2