NAG FL Interface
g01kff (pdf_​gamma)

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1 Purpose

g01kff 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

Fortran Interface
Function g01kff ( x, a, b, ifail)
Real (Kind=nag_wp) :: g01kff
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x, a, b
C Header Interface
#include <nag.h>
double  g01kff_ (const double *x, const double *a, const double *b, Integer *ifail)
The routine may be called by the names g01kff or nagf_stat_pdf_gamma.

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 Real (Kind=nag_wp) Input
On entry: x, the value at which the PDF is to be evaluated.
2: a Real (Kind=nag_wp) Input
On entry: α, the shape parameter of the gamma distribution.
Constraint: a>0.0.
3: b Real (Kind=nag_wp) Input
On entry: β, the scale parameter of the gamma distribution.
  • b>0.0;
  • xb<1x02amf().
4: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
If ifail0, then g01kff returns 0.0.
On entry, a=value.
Constraint: a>0.0.
On entry, b=value.
Constraint: b>0.0.
Computation abandoned owing to overflow due to extreme parameter values.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL 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 FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

g01kff 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 (g01kffe.f90)

10.2 Program Data

Program Data (g01kffe.d)

10.3 Program Results

Program Results (g01kffe.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