nag_rngs_beta (g05lec) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_rngs_beta (g05lec)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_rngs_beta (g05lec) generates a vector of pseudorandom numbers taken from a beta distribution with a and b.

2  Specification

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

3  Description

The beta distribution has PDF (probability density function)
fx = Γa+b ΓaΓb xa-1 1-xb-1, if 0x1;a,b>0.0, 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, Jhnk'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 parameter 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_rngs_init_repeatable (g05kbc) (for a repeatable sequence if computed sequentially) or nag_rngs_init_nonrepeatable (g05kcc) (for a non-repeatable sequence) must be called prior to the first call to nag_rngs_beta (g05lec).

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:     adoubleInput
On entry: a, the parameter of the beta distribution.
Constraint: a>0.0.
2:     bdoubleInput
On entry: b, the parameter of the beta distribution.
Constraint: b>0.0.
3:     nIntegerInput
On entry: n, the number of pseudorandom numbers to be generated.
Constraint: n0.
4:     x[n]doubleOutput
On exit: the n pseudorandom numbers from the specified beta distribution.
5:     igenIntegerInput
On entry: must contain the identification number for the generator to be used to return a pseudorandom number and should remain unchanged following initialization by a prior call to nag_rngs_init_repeatable (g05kbc) or nag_rngs_init_nonrepeatable (g05kcc).
6:     iseed[4]IntegerCommunication Array
On entry: contains values which define the current state of the selected generator.
On exit: contains updated values defining the new state of the selected generator.
7:     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_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_rngs_beta (g05lec) 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_rngs_beta (g05lec), after initialization by nag_rngs_init_repeatable (g05kbc).

9.1  Program Text

Program Text (g05lece.c)

9.2  Program Data

None.

9.3  Program Results

Program Results (g05lece.r)


nag_rngs_beta (g05lec) (PDF version)
g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual

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