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 Vectorized Routines in the G01 Chapter Introduction for further information.

References

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

Parameters

Compulsory Input Parameters

1: $tail (ltail)$ – cell array of strings

Indicates whether a lower or upper tail probability is required. For

j = ((i - 1) mod ltail) + 1

, for

i = 1, 2, \dots, \max (ltail, \lg, la, lb)

$tail (j) ='L'$: The lower tail probability is returned, i.e., $p_{i} = P (G_{i} \leq g_{i} : α_{i}, β_{i})$ .
$tail (j) ='U'$: The upper tail probability is returned, i.e., $p_{i} = P (G_{i} \geq g_{i} : α_{i}, β_{i})$ .

Constraint:

tail (j) ='L'

'U'

, for

j = 1, 2, \dots, ltail

2: $g (\lg)$ – double array

g_{i}

, the value of the gamma variate with

g_{i} = g (j)

j = ((i - 1) mod \lg) + 1

Constraint:

g (j) \geq 0.0

, for

j = 1, 2, \dots, \lg

3: $a (la)$ – double array

The parameter

α_{i}

of the gamma distribution with

α_{i} = a (j)

j = ((i - 1) mod la) + 1

Constraint:

a (j) > 0.0

, for

j = 1, 2, \dots, la

4: $b (lb)$ – double array

The parameter

β_{i}

of the gamma distribution with

β_{i} = b (j)

j = ((i - 1) mod lb) + 1

Constraint:

b (j) > 0.0

, for

j = 1, 2, \dots, lb

Optional Input Parameters

1: $ltail$ – int64int32nag_int scalar: Default: the dimension of the array tail.
The length of the array tail.

Constraint: $ltail > 0$ .
2: $\lg$ – int64int32nag_int scalar: Default: the dimension of the array g.
The length of the array g.

Constraint: $\lg > 0$ .
3: $la$ – int64int32nag_int scalar: Default: the dimension of the array a.
The length of the array a.

Constraint: $la > 0$ .
4: $lb$ – int64int32nag_int scalar: Default: the dimension of the array b.
The length of the array b.

Constraint: $lb > 0$ .

Output Parameters

1: $p (:)$ – double array

The dimension of the array p will be

\max (\lg, la, lb, ltail)

p_{i}

, the probabilities of the beta distribution.

2: $ivalid (:)$ – int64int32nag_int array

The dimension of the array ivalid will be

\max (\lg, la, lb, ltail)

ivalid (i)

indicates any errors with the input arguments, with

$ivalid (i) = 0$

No error.

$ivalid (i) = 1$

On entry,

invalid value supplied in tail when calculating

p_{i}

$ivalid (i) = 2$

On entry,

g_{i} < 0.0

$ivalid (i) = 3$

On entry,	$α_{i} \leq 0.0$ ,
or	$β_{i} \leq 0.0$ .

$ivalid (i) = 4$

The solution did not converge in

600

iterations, see nag_specfun_gamma_incomplete (s14ba) for details. The probability returned should be a reasonable approximation to the solution.

3: $ifail$ – 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:

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

W $ifail = 1$: On entry, at least one value of g, a, b or tail was invalid, or the solution did not converge.
Check ivalid for more information.

$ifail = 2$: Constraint: $ltail > 0$ .

$ifail = 3$: Constraint: $\lg > 0$ .

$ifail = 4$: Constraint: $la > 0$ .

$ifail = 5$: Constraint: $lb > 0$ .

$ifail = - 99$: 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.

Further Comments

The time taken by nag_stat_prob_gamma_vector (g01sf) to calculate each probability varies slightly with the input arguments

g_{i}

α_{i}

and

β_{i}

Example

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

Open in the MATLAB editor: g01sf_example

function g01sf_example


fprintf('g01sf example results\n\n');

tail = {'L'};
g = [15.5; 0.5; 10; 5];
a = [4; 4; 1; 2];
b = [2; 1; 2; 2];
% calculate probability
[p, ivalid, ifail] = g01sf(tail, g, a, b);

fprintf('Gamma deviate    Alpha     Beta    Probability\n');
lg    = numel(g);
la    = numel(a);
lb    = numel(b);
ltail = numel(tail);
len   = max ([lg, la, lb, ltail]);
for i=0:len-1
  fprintf('%9.2f%13.2f%9.2f%10.4f\n', g(mod(i,lg)+1), a(mod(i,la)+1), ...
          b(mod(i,lb)+1), p(i+1));
end

g01sf example results

Gamma deviate    Alpha     Beta    Probability
    15.50         4.00     2.00    0.9499
     0.50         4.00     1.00    0.0018
    10.00         1.00     2.00    0.9933
     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