nag_rand_beta (g05sbc) : NAG Library, Mark 23

3 Description

The beta distribution has PDF (probability density function)

\begin{array}{l} f (x) = \frac{Γ (a + b)}{Γ (a) Γ (b)} x^{a - 1} {(1 - x)}^{b - 1} & if 0 \leq x \leq 1; ​ a, b > 0, \\ f (x) = 0 & otherwise. \end{array}

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 $u_{1}^{1 / a} / (\begin{matrix} u_{1}^{1 / a} + u_{2}^{1 / b} \end{matrix})$ , 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 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: n – IntegerInput

On entry:

n

, the number of pseudorandom numbers to be generated.

Constraint:

n \geq 0

2: a – doubleInput

On entry:

a

, the parameter of the beta distribution.

Constraint:

a > 0.0

3: b – doubleInput

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: fail – NagError *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:

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.

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

NAG Library Function Document

nag_rand_beta (g05sbc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Documentnag_rand_beta (g05sbc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_rand_beta (g05sbc)