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NAG Toolbox: nag_rand_dist_gamma (g05sj)

Purpose

nag_rand_dist_gamma (g05sj) generates a vector of pseudorandom numbers taken from a gamma distribution with parameters aa and bb.

Syntax

[state, x, ifail] = g05sj(n, a, b, state)
[state, x, ifail] = nag_rand_dist_gamma(n, a, b, state)

Description

The gamma distribution has PDF (probability density function)
f(x) = 1/(baΓ(a))xa1ex / b if ​x0;  a,b > 0
f(x) = 0 otherwise.
f(x)= 1baΓ(a) xa-1e-x/b if ​x0;  a,b>0 f(x)=0 otherwise.
One of three algorithms is used to generate the variates depending upon the value of aa:
(i) if a < 1a<1, a switching algorithm described by Dagpunar (1988) (called G6) is used. The target distributions are f1(x) = caxa1 / taf1(x)=caxa-1/ta and f2(x) = (1c)e(xt)f2(x)=(1-c)e-(x-t), where c = t / (t + aet)c=t/(t+ae-t), and the switching parameter, tt, is taken as 1a1-a. This is similar to Ahrens and Dieter's GS algorithm (see Ahrens and Dieter (1974)) in which t = 1t=1;
(ii) if a = 1a=1, the gamma distribution reduces to the exponential distribution and the method based on the logarithmic transformation of a uniform random variate is used;
(iii) if a > 1a>1, the algorithm given by Best (1978) is used. This is based on using a Student's tt-distribution with two degrees of freedom as the target distribution in an envelope rejection method.
One of the initialization functions nag_rand_init_repeat (g05kf) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeat (g05kg) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_dist_gamma (g05sj).

References

Ahrens J H and Dieter U (1974) Computer methods for sampling from gamma, beta, Poisson and binomial distributions Computing 12 223–46
Best D J (1978) Letter to the Editor Appl. Statist. 27 181
Dagpunar J (1988) Principles of Random Variate Generation Oxford University Press
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

Parameters

Compulsory Input Parameters

1:     n – int64int32nag_int scalar
nn, the number of pseudorandom numbers to be generated.
Constraint: n0n0.
2:     a – double scalar
aa, the parameter of the gamma distribution.
Constraint: a > 0.0a>0.0.
3:     b – double scalar
bb, the parameter of the gamma distribution.
Constraint: b > 0.0b>0.0.
4:     state( : :) – int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains information on the selected base generator and its current state.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     state( : :) – int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains updated information on the state of the generator.
2:     x(n) – double array
The nn pseudorandom numbers from the specified gamma distribution.
3:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
On entry, n < 0n<0.
  ifail = 2ifail=2
On entry, a0.0a0.0.
  ifail = 3ifail=3
On entry, b0.0b0.0.
  ifail = 4ifail=4
On entry,state vector was not initialized or has been corrupted.

Accuracy

Not applicable.

Further Comments

None.

Example

function nag_rand_dist_gamma_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
n = int64(5);
a = 5;
b = 1;
% Initialize the generator to a repeatable sequence
[state, ifail] = nag_rand_init_repeat(genid, subid, seed);
[state, x, ifail] = nag_rand_dist_gamma(n, a, b, state)
 

state =

                   17
                 1234
                    1
                    0
                 3990
                  775
                 3088
                31015
                17917
                13895
                19930
                    8
                    0
                 1234
                    1
                    1
                 1234


x =

    5.0702
    6.1337
    3.1018
    3.9863
    4.9648


ifail =

                    0


function g05sj_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
n = int64(5);
a = 5;
b = 1;
% Initialize the generator to a repeatable sequence
[state, ifail] = g05kf(genid, subid, seed);
[state, x, ifail] = g05sj(n, a, b, state)
 

state =

                   17
                 1234
                    1
                    0
                 3990
                  775
                 3088
                31015
                17917
                13895
                19930
                    8
                    0
                 1234
                    1
                    1
                 1234


x =

    5.0702
    6.1337
    3.1018
    3.9863
    4.9648


ifail =

                    0



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

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