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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_stat_pdf_gamma (g01kf)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_stat_pdf_gamma (g01kf) 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.


[result, ifail] = g01kf(x, a, b)
[result, ifail] = nag_stat_pdf_gamma(x, a, b)


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 Further Comments).


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


Compulsory Input Parameters

1:     x – double scalar
x, the value at which the PDF is to be evaluated.
2:     a – double scalar
α, the shape argument of the gamma distribution.
Constraint: a>0.0.
3:     b – double scalar
β, the scale argument of the gamma distribution.
  • b>0.0;
  • xb<1x02am.

Optional Input Parameters


Output Parameters

1:     result – double scalar
The result of the function.
2:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
If ifail0, then nag_stat_pdf_gamma (g01kf) returns 0.0.
Constraint: a>0.0.
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.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


Not applicable.

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)
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 .  


This example prints the value of the gamma distribution PDF at six different points x with differing a and b.
function g01kf_example

fprintf('g01kf example results\n\n');

x      = [0.1, 3, 6,  4,   9,  16];
a      = [3,  10, 5, 10,   9,   3.5];
b      = [2,  11, 1,  0.1, 0.5, 2.5];
result = x;
fprintf('\n      x            a            b          pdf\n');

for i=1:numel(x)
 [result(i), ifail] = g01kf( ...
			     x(i), a(i), b(i));

fprintf('%12.4e %12.4e %12.4e %12.4e\n', [x; a; b; result]);


function g01kf_plot
  fig1 = figure;
  hold on;
  a  = [2, 9];
  b  = [2, 0.5];
  x  = [0:0.1:10];
  alpha = '\alpha';
  beta  = '\beta';
  for i=1:2
    for j=1:numel(x)
      [y{i}(j), ifail] = g01kf( ...
				x(j), a(i), b(i));
    l{i} = sprintf('%s = %3.1f, %s = %3.1f', alpha, a(i), beta, b(i));
  title('Gamma Distributions');
  hold off;
g01kf example results

      x            a            b          pdf
  1.0000e-01   3.0000e+00   2.0000e+00   5.9452e-04
  3.0000e+00   1.0000e+01   1.1000e+01   1.5921e-12
  6.0000e+00   5.0000e+00   1.0000e+00   1.3385e-01
  4.0000e+00   1.0000e+01   1.0000e-01   3.0690e-08
  9.0000e+00   9.0000e+00   5.0000e-01   8.3251e-03
  1.6000e+01   3.5000e+00   2.5000e+00   2.0723e-02

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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