1: $g$ – double scalar: $g$ , the value of the gamma variate.

Constraint: $g \geq 0.0$ .
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.

$tail ='L'$: The lower tail probability is returned, that is $P (G \leq g : α, β)$ .
$tail ='U'$: The upper tail probability is returned, that is $P (G \geq g : α, β)$ .

Constraint:

tail ='L'

'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:

ifail = 1

2

3

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.

$ifail = 1$

On entry,

tail \neq'L'

'U'

$ifail = 2$

On entry,

g < 0.0

$ifail = 3$

On entry,	$a \leq 0.0$ ,
or	$b \leq 0.0$ .

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.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

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

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.

Example

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

Open in the MATLAB editor: g01ef_example

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);
end

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