nag_rand_beta (g05sbc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_rand_beta (g05sbc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_rand_beta (g05sbc) generates a vector of pseudorandom numbers taken from a beta distribution with parameters a and b.

2  Specification

#include <nag.h>
#include <nagg05.h>
void  nag_rand_beta (Integer n, double a, double b, Integer state[], double x[], NagError *fail)

3  Description

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_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_beta (g05sbc).

4  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

5  Arguments

1:     nIntegerInput
On entry: n, the number of pseudorandom numbers to be generated.
Constraint: n0.
2:     adoubleInput
On entry: a, the parameter of the beta distribution.
Constraint: a>0.0.
3:     bdoubleInput
On entry: b, the parameter of the beta distribution.
Constraint: b>0.0.
4:     state[dim]IntegerCommunication Array
Note: the actual argument supplied must be the array state supplied to the initialization functions nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).
On entry: contains information on the selected base generator and its current state.
On exit: contains updated information on the state of the generator.
5:     x[n]doubleOutput
On exit: the n pseudorandom numbers from the specified beta distribution.
6:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

On entry, argument value had an illegal value.
On entry, n=value.
Constraint: n0.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
On entry, state vector has been corrupted or not initialized.
On entry, a=value.
Constraint: a>0.0.
On entry, b=value.
Constraint: b>0.0.

7  Accuracy

Not applicable.

8  Further Comments

To generate an observation, y, from the beta distribution of the second kind from an observation, x, generated by nag_rand_beta (g05sbc) the transformation, y=x/1-x, may be used.

9  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_beta (g05sbc), after initialization by nag_rand_init_repeatable (g05kfc).

9.1  Program Text

Program Text (g05sbce.c)

9.2  Program Data


9.3  Program Results

Program Results (g05sbce.r)

nag_rand_beta (g05sbc) (PDF version)
g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012