nag_gamma_pdf (g01kfc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_gamma_pdf (g01kfc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

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

2  Specification

#include <nag.h>
#include <nagg01.h>
double  nag_gamma_pdf (double x, double a, double b, NagError *fail)

3  Description

The gamma distribution has PDF
fx= 1βαΓα xα-1e-x/β if ​x0;  α,β>0 fx=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 8).

4  References

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

5  Arguments

1:     xdoubleInput
On entry: x, the value at which the PDF is to be evaluated.
2:     adoubleInput
On entry: α, the shape argument of the gamma distribution.
Constraint: a>0.0.
3:     bdoubleInput
On entry: β, the scale argument of the gamma distribution.
Constraints:
  • b>0.0;
  • xb<1nag_real_safe_small_number.
4:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_INTERNAL_ERROR
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.
NE_OVERFLOW
Computation abandoned owing to overflow due to extreme parameter values.
NE_REAL
On entry, a=value.
Constraint: a>0.0.
On entry, b=value.
Constraint: b>0.0.

7  Accuracy

Not applicable.

8  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,
px;λ = λx x! e-λ . (1)
The usual way of computing this quantity would be to take the logarithm and calculate,
logx;λ = x logλ - log x! - λ .
For large x and λ, xlogλ and logx! 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, logx!2.7×107 and logpx;λ=-8.17326744645834. But calculated with the method shown later we have logpx;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 - Dx;λ , (2)
where Dx;λ, the deviance for the Poisson distribution is given by,
Dx;λ = log p x;x - log p x;λ , = λ D0 x λ , (3)
and
D0 ε = ε logε + 1 - ε .
For ε close to 1, D0ε 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 D0ε can be evaluated directly. In addition, Loader suggests evaluating logx! using the Stirling–De Moivre series,
logx! = 12 log 2πx + x logx -x + δx , (4)
where the error δx is given by
δx = 112x - 1 360x3 + 1 1260x5 + O x-7 .
Finally logpx;λ can be evaluated by combining equations (1)(4) to get,
p x;λ = 1 2πx e - δx - λ D0 x/λ .

9  Example

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

9.1  Program Text

Program Text (g01kfce.c)

9.2  Program Data

Program Data (g01kfce.d)

9.3  Program Results

Program Results (g01kfce.r)

Produced by GNUPLOT 4.4 patchlevel 0 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 a=2, b=2 a=9, b=0.5

nag_gamma_pdf (g01kfc) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012