(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 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 g05kfc (for a repeatable sequence if computed sequentially) or g05kgc (for a non-repeatable sequence) must be called prior to the first call to 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: $n$ – Integer Input: On entry: $n$ , the number of pseudorandom numbers to be generated.

Constraint: $n \geq 0$ .
2: $a$ – double Input: On entry: $a$ , the parameter of the beta distribution.

Constraint: $a > 0.0$ .
3: $b$ – double Input: On entry: $b$ , the parameter of the beta distribution.

Constraint: $b > 0.0$ .
4: $state [\dim]$ – Integer Communication Array: Note: the dimension, $\dim$ , of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to 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: $x [n]$ – double Output: On exit: the $n$ pseudorandom numbers from the specified beta distribution.
6: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_INVALID_STATE: On entry, state vector has been corrupted or not initialized.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL: On entry, $a = 〈value〉$ .
Constraint: $a > 0.0$ .

On entry, $b = 〈value〉$ .
Constraint: $b > 0.0$ .

7 Accuracy

Not applicable.

8 Parallelism and Performance

g05sbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

To generate an observation,

y

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

x

, generated by g05sbc the transformation,

y = x / (1 - x)

, may be used.

10 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 g05sbc, after initialization by g05kfc.

10.1 Program Text

Program Text (g05sbce.c)

10.2 Program Data

None.

10.3 Program Results

Program Results (g05sbce.r)

NAG Library Manual, Mark 27

Interfaces: FL CL AD

NAG CL Interface Introduction

G05 (Rand) Chapter Contents

G05 (Rand) Chapter Introduction

g05sb: FL CL AD

NAG CL Interfaceg05sbc (dist_​beta)

▸▿ Contents

1 Purpose

2 Specification

3 Description