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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_rand_dist_beta (g05sb)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_rand_dist_beta (g05sb) generates a vector of pseudorandom numbers taken from a beta distribution with parameters a and b.


[state, x, ifail] = g05sb(n, a, b, state)
[state, x, ifail] = nag_rand_dist_beta(n, a, b, state)


The beta distribution has PDF (probability density function)
fx = Γa+b Γa Γb xa-1 1-x b-1 if  0x1 ; ​ a,b>0 , fx=0 otherwise.  
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 u11/a/ u11/a+u21/b , where u1 and u2 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 f1x=βxβ and f2x=α1-xβ-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).


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


Compulsory Input Parameters

1:     n int64int32nag_int scalar
n, the number of pseudorandom numbers to be generated.
Constraint: n0.
2:     a – double scalar
a, the parameter of the beta distribution.
Constraint: a>0.0.
3:     b – double scalar
b, the parameter of the beta distribution.
Constraint: b>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


Output Parameters

1:     state: int64int32nag_int array
Contains updated information on the state of the generator.
2:     xn – double array
The n pseudorandom numbers from the specified beta distribution.
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:
Constraint: n0.
Constraint: a>0.0.
Constraint: b>0.0.
On entry, state vector has been corrupted or not initialized.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


Not applicable.

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.


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);


g05sb example results


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

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