g05sj:: Random Number Generators (NAG Toolbox)

Description

The gamma distribution has PDF (probability density function)

\begin{array}{l} f (x) = \frac{1}{b^{a} Γ (a)} x^{a - 1} e^{- x / b} & if ​ x \geq 0; a, b > 0 \\ f (x) = 0 & otherwise. \end{array}

One of three algorithms is used to generate the variates depending upon the value of

a

(i)	if $a < 1$ , a switching algorithm described by Dagpunar (1988) (called G6) is used. The target distributions are $f_{1} (x) = c a x^{a - 1} / t^{a}$ and $f_{2} (x) = (1 - c) e^{- (x - t)}$ , where $c = t / (t + a e^{- t})$ , and the switching argument, $t$ , is taken as $1 - a$ . This is similar to Ahrens and Dieter's GS algorithm (see Ahrens and Dieter (1974)) in which $t = 1$ ;
(ii)	if $a = 1$ , 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 $a > 1$ , the algorithm given by Best (1978) is used. This is based on using a Student's $t$ -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

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Errors or warnings detected by the function:

Accuracy

Further Comments

Example

This example prints a set of five pseudorandom numbers from a gamma distribution with parameters

a = 5.0

and

b = 1.0

, generated by a single call to nag_rand_dist_gamma (g05sj), after initialization by nag_rand_init_repeat (g05kf).

function g05sj_example


fprintf('g05sj example results\n\n');

% Initialize the base generator to a repeatable sequence
seed  = [int64(1762543)];
genid = int64(1);
subid = int64(1);
[state, ifail] = g05kf( ...
                        genid, subid, seed);

% Number of variates
n = int64(5);

% Parameters
a = 5;
b = 1;

% Generate variates from Gamma distribution
[state, x, ifail] = g05sj( ...
                           n, a, b, state);

disp('Variates');
disp(x);

NAG Toolbox: nag_rand_dist_gamma (g05sj)

▸▿ Contents

Purpose

Syntax