g05sb:: Random Number Generators (NAG Toolbox)

One of four algorithms is used to generate the variates depending on the values of

a

and

b

. Let

α

be the maximum and

β

be the minimum of

a

and

b

. Then the algorithms are as follows:

(i)	if $α < 0.5$ , Johnk's algorithm is used, see for example Dagpunar (1988). This generates the beta variate as $u_{1}^{1 / a} / (\begin{matrix} u_{1}^{1 / a} + u_{2}^{1 / b} \end{matrix})$ , where $u_{1}$ and $u_{2}$ are uniformly distributed random variates;
(ii)	if $β > 1$ , the algorithm BB given by Cheng (1978) is used. This involves the generation of an observation from a beta distribution of the second kind by the envelope rejection method using a log-logistic target distribution and then transforming it to a beta variate;
(iii)	if $α > 1$ and $β < 1$ , the switching algorithm given by Atkinson (1979) is used. The two target distributions used are $f_{1} (x) = β x^{β}$ and $f_{2} (x) = α {(1 - x)}^{β - 1}$ , along with the approximation to the switching argument of $t = (1 - β) / (α + 1 - β)$ ;
(iv)	in all other cases, Cheng's BC algorithm (see Cheng (1978)) is used with modifications suggested by Dagpunar (1988). This algorithm is similar to BB, used when $β > 1$ , but is tuned for small values of $a$ and $b$ .

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_beta (g05sb).

References

Atkinson A C (1979) A family of switching algorithms for the computer generation of beta random variates Biometrika 66 141–5

Cheng R C H (1978) Generating beta variates with nonintegral shape parameters Comm. ACM 21 317–322

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

To generate an observation,

y

, from the beta distribution of the second kind from an observation,

x

, generated by nag_rand_dist_beta (g05sb) the transformation,

y = x / (1 - x)

, may be used.

Example

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

a = 2.0

and

b = 2.0

, generated by a single call to nag_rand_dist_beta (g05sb), after initialization by nag_rand_init_repeat (g05kf).

function g05sb_example


fprintf('g05sb 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 = 2;
b = 2;

% Generate variates from beta distribution
[state, x, ifail] = g05sb( ...
                           n, a, b, state);

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

NAG Toolbox: nag_rand_dist_beta (g05sb)

▸▿ Contents

Purpose

Syntax

Description