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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_stat_prob_gamma (g01ef)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_stat_prob_gamma (g01ef) returns the lower or upper tail probability of the gamma distribution, with parameters α and β.


[result, ifail] = g01ef(g, a, b, 'tail', tail)
[result, ifail] = nag_stat_prob_gamma(g, a, b, 'tail', tail)
Note: the interface to this routine has changed since earlier releases of the toolbox:
At Mark 23: tail was made optional (default 'L')


The lower tail probability for the gamma distribution with parameters α and β, PGg, is defined by:
P Gg ; α,β = 1 βα Γα 0g Gα-1 e-G/β dG ,   α>0.0 , ​ β>0.0 .  
The mean of the distribution is αβ and its variance is αβ2. The transformation Z=Gβ is applied to yield the following incomplete gamma function in normalized form,
P Gg ; α ,β = P Zg/β : α,1.0 = 1 Γα 0g/β Zα-1 e-Z dZ .  
This is then evaluated using nag_specfun_gamma_incomplete (s14ba).


Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth


Compulsory Input Parameters

1:     g – double scalar
g, the value of the gamma variate.
Constraint: g0.0.
2:     a – double scalar
The parameter α of the gamma distribution.
Constraint: a>0.0.
3:     b – double scalar
The parameter β of the gamma distribution.
Constraint: b>0.0.

Optional Input Parameters

1:     tail – string (length ≥ 1)
Default: 'L'
Indicates whether an upper or lower tail probability is required.
The lower tail probability is returned, that is PGg:α,β.
The upper tail probability is returned, that is PGg:α,β.
Constraint: tail='L' or 'U'.

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 ifail=1, 2, 3 or 4 on exit, then nag_stat_prob_gamma (g01ef) returns 0.0.

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

On entry,tail'L' or 'U'.
On entry,g<0.0.
On entry,a0.0,
W  ifail=4
The solution did not converge in 600 iterations. See nag_specfun_gamma_incomplete (s14ba). The probability returned should be a reasonable approximation to the solution.
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.


The result should have a relative accuracy of machine precision. There are rare occasions when the relative accuracy attained is somewhat less than machine precision but the error should not exceed more than 1 or 2 decimal places. Note also that there is a limit of 18 decimal places on the achievable accuracy, because constants in nag_specfun_gamma_incomplete (s14ba) are given to this precision.

Further Comments

The time taken by nag_stat_prob_gamma (g01ef) varies slightly with the input arguments g, a and b.


This example reads in values from a number of gamma distributions and computes the associated lower tail probabilities.
function g01ef_example

fprintf('g01ef example results\n\n');

% Lower tail probabilities for Gamma distribution
g    = [   15.5;     0.5;    10.0;     5.0];
a    = [    4.0;     4.0;     1.0;     2.0];
b    = [    2.0;     1.0;     2.0;     2.0];
tail = {'Lower'; 'Lower'; 'Lower'; 'Lower'};

fprintf('  Tail    G      a      b     probability\n');
for j = 1:numel(g);

  [p, ifail] = g01ef( ...
                      g(j), a(j), b(j), 'tail', tail{j});

  fprintf('%4s%8.2f%8.2f%8.2f%12.4f\n', tail{j}(1), g(j), a(j), b(j), p);

g01ef example results

  Tail    G      a      b     probability
   L   15.50    4.00    2.00      0.9499
   L    0.50    4.00    1.00      0.0018
   L   10.00    1.00    2.00      0.9933
   L    5.00    2.00    2.00      0.7127

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

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