nag_rand_beta (g05sbc) (PDF version)
g05 Chapter Contents
g05 Chapter Introduction
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

NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
NE_INTERNAL_ERROR
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.
NE_INVALID_STATE
On entry, state vector has been corrupted or not initialized.
NE_REAL
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

None.

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