NAG CL Interface
g01kkc (pdf_​gamma_​vector)

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1 Purpose

g01kkc returns a number of values of the probability density function (PDF), or its logarithm, for the gamma distribution.

2 Specification

#include <nag.h>
void  g01kkc (Nag_Boolean ilog, Integer lx, const double x[], Integer la, const double a[], Integer lb, const double b[], double pdf[], Integer ivalid[], NagError *fail)
The function may be called by the names: g01kkc, nag_stat_pdf_gamma_vector or nag_gamma_pdf_vector.

3 Description

The gamma distribution with shape parameter αi and scale parameter βi has PDF
f (xi,αi,βi) = 1 βi αi Γ(αi) xi αi-1 e -xi / βi if ​ xi 0 ;   αi , βi > 0 f(xi,αi,βi)=0 otherwise.  
If 0.01xi,αi,βi100 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).
The input arrays to this function are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section 2.6 in the G01 Chapter Introduction for further information.

4 References

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

5 Arguments

1: ilog Nag_Boolean Input
On entry: the value of ilog determines whether the logarithmic value is returned in pdf.
f(xi,αi,βi), the probability density function is returned.
log(f(xi,αi,βi)), the logarithm of the probability density function is returned.
2: lx Integer Input
On entry: the length of the array x.
Constraint: lx>0.
3: x[lx] const double Input
On entry: xi, the values at which the PDF is to be evaluated with xi=x[j], j=(i-1) mod lx, for i=1,2,,max(lx,la,lb).
4: la Integer Input
On entry: the length of the array a.
Constraint: la>0.
5: a[la] const double Input
On entry: αi, the shape parameter with αi=a[j], j=(i-1) mod la.
Constraint: a[j-1]>0.0, for j=1,2,,la.
6: lb Integer Input
On entry: the length of the array b.
Constraint: lb>0.
7: b[lb] const double Input
On entry: βi, the scale parameter with βi=b[j], j=(i-1) mod lb.
Constraint: b[j-1]>0.0, for j=1,2,,lb.
8: pdf[dim] double Output
Note: the dimension, dim, of the array pdf must be at least max(lx,la,lb).
On exit: f(xi,αi,βi) or log(f(xi,αi,βi)).
9: ivalid[dim] Integer Output
Note: the dimension, dim, of the array ivalid must be at least max(lx,la,lb).
On exit: ivalid[i-1] indicates any errors with the input arguments, with
No error.
xiβi overflows, the value returned should be a reasonable approximation.
10: 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

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, array size=value.
Constraint: la>0.
On entry, array size=value.
Constraint: lb>0.
On entry, array size=value.
Constraint: lx>0.
On entry, argument value had an illegal value.
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.
On entry, at least one value of x, a or b was invalid.
Check ivalid for more information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

g01kkc 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(p(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 xi with differing αi and βi.

10.1 Program Text

Program Text (g01kkce.c)

10.2 Program Data

Program Data (g01kkce.d)

10.3 Program Results

Program Results (g01kkce.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